/
l3int.dtx
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l3int.dtx
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% \iffalse meta-comment
%
%% File: l3int.dtx
%
% Copyright (C) 1990-2020 The LaTeX3 Project
%
% It may be distributed and/or modified under the conditions of the
% LaTeX Project Public License (LPPL), either version 1.3c of this
% license or (at your option) any later version. The latest version
% of this license is in the file
%
% https://www.latex-project.org/lppl.txt
%
% This file is part of the "l3kernel bundle" (The Work in LPPL)
% and all files in that bundle must be distributed together.
%
% -----------------------------------------------------------------------
%
% The development version of the bundle can be found at
%
% https://github.com/latex3/latex3
%
% for those people who are interested.
%
%<*driver>
\documentclass[full,kernel]{l3doc}
\begin{document}
\DocInput{\jobname.dtx}
\end{document}
%</driver>
% \fi
%
% \title{^^A
% The \pkg{l3int} package\\ Integers^^A
% }
%
% \author{^^A
% The \LaTeX3 Project\thanks
% {^^A
% E-mail:
% \href{mailto:latex-team@latex-project.org}
% {latex-team@latex-project.org}^^A
% }^^A
% }
%
% \date{Released 2020-01-31}
%
% \maketitle
%
% \begin{documentation}
%
% Calculation and comparison of integer values can be carried out
% using literal numbers, \texttt{int} registers, constants and
% integers stored in token list variables. The standard operators
% \texttt{+}, \texttt{-}, \texttt{/} and \texttt{*} and
% parentheses can be used within such expressions to carry
% arithmetic operations. This module carries out these functions
% on \emph{integer expressions} (\enquote{\texttt{intexpr}}).
%
% \section{Integer expressions}
%
% \begin{function}[EXP]{\int_eval:n}
% \begin{syntax}
% \cs{int_eval:n} \Arg{integer expression}
% \end{syntax}
% Evaluates the \meta{integer expression} and leaves the result in the
% input stream as an integer denotation: for positive results an
% explicit sequence of decimal digits not starting with~\texttt{0},
% for negative results \texttt{-}~followed by such a sequence, and
% \texttt{0}~for zero. The \meta{integer expression} should consist,
% after expansion, of \texttt{+}, \texttt{-}, \texttt{*}, \texttt{/},
% \texttt{(}, \texttt{)} and of course integer operands. The result
% is calculated by applying standard mathematical rules with the
% following peculiarities:
% \begin{itemize}
% \item \texttt{/} denotes division rounded to the closest integer with
% ties rounded away from zero;
% \item there is an error and the overall expression evaluates to zero
% whenever the absolute value of any intermediate result exceeds
% $2^{31}-1$, except in the case of scaling operations
% $a$\texttt{*}$b$\texttt{/}$c$, for which $a$\texttt{*}$b$ may be
% arbitrarily large;
% \item parentheses may not appear after unary \texttt{+} or
% \texttt{-}, namely placing \texttt{+(} or \texttt{-(} at the start
% of an expression or after \texttt{+}, \texttt{-}, \texttt{*},
% \texttt{/} or~\texttt{(} leads to an error.
% \end{itemize}
% Each integer operand can be either an integer variable (with no need
% for \cs{int_use:N}) or an integer denotation. For example both
% \begin{verbatim}
% \int_eval:n { 5 + 4 * 3 - ( 3 + 4 * 5 ) }
% \end{verbatim}
% and
% \begin{verbatim}
% \tl_new:N \l_my_tl
% \tl_set:Nn \l_my_tl { 5 }
% \int_new:N \l_my_int
% \int_set:Nn \l_my_int { 4 }
% \int_eval:n { \l_my_tl + \l_my_int * 3 - ( 3 + 4 * 5 ) }
% \end{verbatim}
% evaluate to $-6$ because \cs[no-index]{l_my_tl} expands to the
% integer denotation~|5|. As the \meta{integer expression} is fully
% expanded from left to right during evaluation, fully expandable and
% restricted-expandable functions can both be used, and \cs{exp_not:n}
% and its variants have no effect while \cs{exp_not:N} may incorrectly
% interrupt the expression.
% \begin{texnote}
% Exactly two expansions are needed to evaluate \cs{int_eval:n}.
% The result is \emph{not} an \meta{internal integer}, and therefore
% requires suitable termination if used in a \TeX{}-style integer
% assignment.
%
% As all \TeX{} integers, integer operands can also be dimension or
% skip variables, converted to integers in~\texttt{sp}, or octal
% numbers given as \texttt{'} followed by digits other than
% \texttt{8} and \texttt{9}, or hexadecimal numbers given as
% |"| followed by digits or upper case letters from
% \texttt{A} to~\texttt{F}, or the character code of some character
% or one-character control sequence, given as \texttt{`}\meta{char}.
% \end{texnote}
% \end{function}
%
% \begin{function}[EXP, added = 2018-03-30]{\int_eval:w}
% \begin{syntax}
% \cs{int_eval:w} \meta{integer expression}
% \end{syntax}
% Evaluates the \meta{integer expression} as described for
% \cs{int_eval:n}. The end of the expression is the first token
% encountered that cannot form part of such an expression. If that
% token is \cs{scan_stop:} it is removed, otherwise not. Spaces do
% \emph{not} terminate the expression. However, spaces terminate
% explict integers, and this may terminate the expression: for
% instance, \cs{int_eval:w} \verb*|1 + 1 9| expands to \texttt{29}
% since the digit~\texttt{9} is not part of the expression.
% \end{function}
%
% \begin{function}[EXP, added = 2018-11-03]{\int_sign:n}
% \begin{syntax}
% \cs{int_sign:n} \Arg{intexpr}
% \end{syntax}
% Evaluates the \meta{integer expression} then leaves $1$ or $0$ or
% $-1$ in the input stream according to the sign of the result.
% \end{function}
%
% \begin{function}[EXP, updated = 2012-09-26]{\int_abs:n}
% \begin{syntax}
% \cs{int_abs:n} \Arg{integer expression}
% \end{syntax}
% Evaluates the \meta{integer expression} as described for
% \cs{int_eval:n} and leaves the absolute value of the result in
% the input stream as an \meta{integer denotation} after two
% expansions.
% \end{function}
%
% \begin{function}[EXP, updated = 2012-09-26]{\int_div_round:nn}
% \begin{syntax}
% \cs{int_div_round:nn} \Arg{intexpr_1} \Arg{intexpr_2}
% \end{syntax}
% Evaluates the two \meta{integer expressions} as described earlier,
% then divides the first value by the second, and rounds the result
% to the closest integer. Ties are rounded away from zero.
% Note that this is identical to using
% |/| directly in an \meta{integer expression}. The result is left in
% the input stream as an \meta{integer denotation} after two expansions.
% \end{function}
%
% \begin{function}[EXP, updated = 2012-02-09]{\int_div_truncate:nn}
% \begin{syntax}
% \cs{int_div_truncate:nn} \Arg{intexpr_1} \Arg{intexpr_2}
% \end{syntax}
% Evaluates the two \meta{integer expressions} as described earlier,
% then divides the first value by the second, and rounds the result
% towards zero. Note that division using |/|
% rounds to the closest integer instead.
% The result is left in the input stream as an
% \meta{integer denotation} after two expansions.
% \end{function}
%
% \begin{function}[EXP, updated = 2012-09-26]{\int_max:nn, \int_min:nn}
% \begin{syntax}
% \cs{int_max:nn} \Arg{intexpr_1} \Arg{intexpr_2}
% \cs{int_min:nn} \Arg{intexpr_1} \Arg{intexpr_2}
% \end{syntax}
% Evaluates the \meta{integer expressions} as described for
% \cs{int_eval:n} and leaves either the larger or smaller value
% in the input stream as an \meta{integer denotation} after two
% expansions.
% \end{function}
%
% \begin{function}[EXP, updated = 2012-09-26]{\int_mod:nn}
% \begin{syntax}
% \cs{int_mod:nn} \Arg{intexpr_1} \Arg{intexpr_2}
% \end{syntax}
% Evaluates the two \meta{integer expressions} as described earlier,
% then calculates the integer remainder of dividing the first
% expression by the second. This is obtained by subtracting
% \cs{int_div_truncate:nn} \Arg{intexpr_1} \Arg{intexpr_2} times
% \meta{intexpr_2} from \meta{intexpr_1}. Thus, the result has the
% same sign as \meta{intexpr_1} and its absolute value is strictly
% less than that of \meta{intexpr_2}. The result is left in the input
% stream as an \meta{integer denotation} after two expansions.
% \end{function}
%
% \section{Creating and initialising integers}
%
% \begin{function}{\int_new:N, \int_new:c}
% \begin{syntax}
% \cs{int_new:N} \meta{integer}
% \end{syntax}
% Creates a new \meta{integer} or raises an error if the name is
% already taken. The declaration is global. The \meta{integer} is
% initially equal to $0$.
% \end{function}
%
% \begin{function}[updated = 2011-10-22]{\int_const:Nn, \int_const:cn}
% \begin{syntax}
% \cs{int_const:Nn} \meta{integer} \Arg{integer expression}
% \end{syntax}
% Creates a new constant \meta{integer} or raises an error if the name
% is already taken. The value of the \meta{integer} is set
% globally to the \meta{integer expression}.
% \end{function}
%
% \begin{function}{\int_zero:N, \int_zero:c, \int_gzero:N, \int_gzero:c}
% \begin{syntax}
% \cs{int_zero:N} \meta{integer}
% \end{syntax}
% Sets \meta{integer} to $0$.
% \end{function}
%
% \begin{function}[added = 2011-12-13]
% {\int_zero_new:N, \int_zero_new:c, \int_gzero_new:N, \int_gzero_new:c}
% \begin{syntax}
% \cs{int_zero_new:N} \meta{integer}
% \end{syntax}
% Ensures that the \meta{integer} exists globally by applying
% \cs{int_new:N} if necessary, then applies
% \cs[index=int_zero:N]{int_(g)zero:N} to leave
% the \meta{integer} set to zero.
% \end{function}
%
% \begin{function}
% {
% \int_set_eq:NN, \int_set_eq:cN, \int_set_eq:Nc, \int_set_eq:cc,
% \int_gset_eq:NN, \int_gset_eq:cN, \int_gset_eq:Nc, \int_gset_eq:cc
% }
% \begin{syntax}
% \cs{int_set_eq:NN} \meta{integer_1} \meta{integer_2}
% \end{syntax}
% Sets the content of \meta{integer_1} equal to that of
% \meta{integer_2}.
% \end{function}
%
% \begin{function}[EXP, pTF, added=2012-03-03]
% {\int_if_exist:N, \int_if_exist:c}
% \begin{syntax}
% \cs{int_if_exist_p:N} \meta{int}
% \cs{int_if_exist:NTF} \meta{int} \Arg{true code} \Arg{false code}
% \end{syntax}
% Tests whether the \meta{int} is currently defined. This does not
% check that the \meta{int} really is an integer variable.
% \end{function}
%
% \section{Setting and incrementing integers}
%
% \begin{function}[updated = 2011-10-22]
% {\int_add:Nn, \int_add:cn, \int_gadd:Nn, \int_gadd:cn}
% \begin{syntax}
% \cs{int_add:Nn} \meta{integer} \Arg{integer expression}
% \end{syntax}
% Adds the result of the \meta{integer expression} to the current
% content of the \meta{integer}.
% \end{function}
%
% \begin{function}{\int_decr:N, \int_decr:c, \int_gdecr:N, \int_gdecr:c}
% \begin{syntax}
% \cs{int_decr:N} \meta{integer}
% \end{syntax}
% Decreases the value stored in \meta{integer} by $1$.
% \end{function}
%
% \begin{function}{\int_incr:N, \int_incr:c, \int_gincr:N, \int_gincr:c}
% \begin{syntax}
% \cs{int_incr:N} \meta{integer}
% \end{syntax}
% Increases the value stored in \meta{integer} by $1$.
% \end{function}
%
% \begin{function}[updated = 2011-10-22]
% {\int_set:Nn, \int_set:cn, \int_gset:Nn, \int_gset:cn}
% \begin{syntax}
% \cs{int_set:Nn} \meta{integer} \Arg{integer expression}
% \end{syntax}
% Sets \meta{integer} to the value of \meta{integer expression},
% which must evaluate to an integer (as described for
% \cs{int_eval:n}).
% \end{function}
%
% \begin{function}[updated = 2011-10-22]
% {\int_sub:Nn, \int_sub:cn, \int_gsub:Nn, \int_gsub:cn}
% \begin{syntax}
% \cs{int_sub:Nn} \meta{integer} \Arg{integer expression}
% \end{syntax}
% Subtracts the result of the \meta{integer expression} from the
% current content of the \meta{integer}.
% \end{function}
%
% \section{Using integers}
%
% \begin{function}[updated = 2011-10-22, EXP]{\int_use:N, \int_use:c}
% \begin{syntax}
% \cs{int_use:N} \meta{integer}
% \end{syntax}
% Recovers the content of an \meta{integer} and places it directly
% in the input stream. An error is raised if the variable does
% not exist or if it is invalid. Can be omitted in places where an
% \meta{integer} is required (such as in the first and third arguments
% of \cs{int_compare:nNnTF}).
% \begin{texnote}
% \cs{int_use:N} is the \TeX{} primitive \tn{the}: this is one of
% several \LaTeX3 names for this primitive.
% \end{texnote}
% \end{function}
%
% \section{Integer expression conditionals}
%
% \begin{function}[EXP,pTF]{\int_compare:nNn}
% \begin{syntax}
% \cs{int_compare_p:nNn} \Arg{intexpr_1} \meta{relation} \Arg{intexpr_2} \\
% \cs{int_compare:nNnTF}
% ~~\Arg{intexpr_1} \meta{relation} \Arg{intexpr_2}
% ~~\Arg{true code} \Arg{false code}
% \end{syntax}
% This function first evaluates each of the \meta{integer expressions}
% as described for \cs{int_eval:n}. The two results are then
% compared using the \meta{relation}:
% \begin{center}
% \begin{tabular}{ll}
% Equal & |=| \\
% Greater than & |>| \\
% Less than & |<| \\
% \end{tabular}
% \end{center}
% This function is less flexible than \cs{int_compare:nTF} but around
% $5$~times faster.
% \end{function}
%
% \begin{function}[updated = 2013-01-13, EXP, pTF]{\int_compare:n}
% \begin{syntax}
% \cs{int_compare_p:n} \\
% ~~\{ \\
% ~~~~\meta{intexpr_1} \meta{relation_1} \\
% ~~~~\ldots{} \\
% ~~~~\meta{intexpr_N} \meta{relation_N} \\
% ~~~~\meta{intexpr_{N+1}} \\
% ~~\} \\
% \cs{int_compare:nTF}
% ~~\{ \\
% ~~~~\meta{intexpr_1} \meta{relation_1} \\
% ~~~~\ldots{} \\
% ~~~~\meta{intexpr_N} \meta{relation_N} \\
% ~~~~\meta{intexpr_{N+1}} \\
% ~~\} \\
% ~~\Arg{true code} \Arg{false code}
% \end{syntax}
% This function evaluates the \meta{integer expressions} as described
% for \cs{int_eval:n} and compares consecutive result using the
% corresponding \meta{relation}, namely it compares \meta{intexpr_1}
% and \meta{intexpr_2} using the \meta{relation_1}, then
% \meta{intexpr_2} and \meta{intexpr_3} using the \meta{relation_2},
% until finally comparing \meta{intexpr_N} and \meta{intexpr_{N+1}}
% using the \meta{relation_N}. The test yields \texttt{true} if all
% comparisons are \texttt{true}. Each \meta{integer expression} is
% evaluated only once, and the evaluation is lazy, in the sense that
% if one comparison is \texttt{false}, then no other \meta{integer
% expression} is evaluated and no other comparison is performed.
% The \meta{relations} can be any of the following:
% \begin{center}
% \begin{tabular}{ll}
% Equal & |=| or |==| \\
% Greater than or equal to & |>=| \\
% Greater than & |>| \\
% Less than or equal to & |<=| \\
% Less than & |<| \\
% Not equal & |!=| \\
% \end{tabular}
% \end{center}
% This function is more flexible than \cs{int_compare:nNnTF} but
% around $5$~times slower.
% \end{function}
%
% \begin{function}[added = 2013-07-24, EXP, noTF]{\int_case:nn}
% \begin{syntax}
% \cs{int_case:nnTF} \Arg{test integer expression} \\
% ~~|{| \\
% ~~~~\Arg{intexpr case_1} \Arg{code case_1} \\
% ~~~~\Arg{intexpr case_2} \Arg{code case_2} \\
% ~~~~\ldots \\
% ~~~~\Arg{intexpr case_n} \Arg{code case_n} \\
% ~~|}| \\
% ~~\Arg{true code}
% ~~\Arg{false code}
% \end{syntax}
% This function evaluates the \meta{test integer expression} and
% compares this in turn to each of the
% \meta{integer expression cases}. If the two are equal then the
% associated \meta{code} is left in the input stream
% and other cases are discarded. If any of the
% cases are matched, the \meta{true code} is also inserted into the
% input stream (after the code for the appropriate case), while if none
% match then the \meta{false code} is inserted. The function
% \cs{int_case:nn}, which does nothing if there is no match, is also
% available. For example
% \begin{verbatim}
% \int_case:nnF
% { 2 * 5 }
% {
% { 5 } { Small }
% { 4 + 6 } { Medium }
% { -2 * 10 } { Negative }
% }
% { No idea! }
% \end{verbatim}
% leaves \enquote{\texttt{Medium}} in the input stream.
% \end{function}
%
% \begin{function}[EXP,pTF]{\int_if_even:n, \int_if_odd:n}
% \begin{syntax}
% \cs{int_if_odd_p:n} \Arg{integer expression}
% \cs{int_if_odd:nTF} \Arg{integer expression}
% ~~\Arg{true code} \Arg{false code}
% \end{syntax}
% This function first evaluates the \meta{integer expression}
% as described for \cs{int_eval:n}. It then evaluates if this
% is odd or even, as appropriate.
% \end{function}
%
% \section{Integer expression loops}
%
% \begin{function}[rEXP]{\int_do_until:nNnn}
% \begin{syntax}
% \cs{int_do_until:nNnn} \Arg{intexpr_1} \meta{relation} \Arg{intexpr_2} \Arg{code}
% \end{syntax}
% Places the \meta{code} in the input stream for \TeX{} to process, and
% then evaluates the relationship between the two
% \meta{integer expressions} as described for \cs{int_compare:nNnTF}.
% If the test is \texttt{false} then the \meta{code} is inserted
% into the input stream again and a loop occurs until the
% \meta{relation} is \texttt{true}.
% \end{function}
%
% \begin{function}[rEXP]{\int_do_while:nNnn}
% \begin{syntax}
% \cs{int_do_while:nNnn} \Arg{intexpr_1} \meta{relation} \Arg{intexpr_2} \Arg{code}
% \end{syntax}
% Places the \meta{code} in the input stream for \TeX{} to process, and
% then evaluates the relationship between the two
% \meta{integer expressions} as described for \cs{int_compare:nNnTF}.
% If the test is \texttt{true} then the \meta{code} is inserted
% into the input stream again and a loop occurs until the
% \meta{relation} is \texttt{false}.
% \end{function}
%
% \begin{function}[rEXP]{\int_until_do:nNnn}
% \begin{syntax}
% \cs{int_until_do:nNnn} \Arg{intexpr_1} \meta{relation} \Arg{intexpr_2} \Arg{code}
% \end{syntax}
% Evaluates the relationship between the two \meta{integer expressions}
% as described for \cs{int_compare:nNnTF}, and then places the
% \meta{code} in the input stream if the \meta{relation} is
% \texttt{false}. After the \meta{code} has been processed by \TeX{} the
% test is repeated, and a loop occurs until the test is
% \texttt{true}.
% \end{function}
%
% \begin{function}[rEXP]{\int_while_do:nNnn}
% \begin{syntax}
% \cs{int_while_do:nNnn} \Arg{intexpr_1} \meta{relation} \Arg{intexpr_2} \Arg{code}
% \end{syntax}
% Evaluates the relationship between the two \meta{integer expressions}
% as described for \cs{int_compare:nNnTF}, and then places the
% \meta{code} in the input stream if the \meta{relation} is
% \texttt{true}. After the \meta{code} has been processed by \TeX{} the
% test is repeated, and a loop occurs until the test is
% \texttt{false}.
% \end{function}
%
% \begin{function}[updated = 2013-01-13, rEXP]{\int_do_until:nn}
% \begin{syntax}
% \cs{int_do_until:nn} \Arg{integer relation} \Arg{code}
% \end{syntax}
% Places the \meta{code} in the input stream for \TeX{} to process, and
% then evaluates the \meta{integer relation}
% as described for \cs{int_compare:nTF}.
% If the test is \texttt{false} then the \meta{code} is inserted
% into the input stream again and a loop occurs until the
% \meta{relation} is \texttt{true}.
% \end{function}
%
% \begin{function}[updated = 2013-01-13, rEXP]{\int_do_while:nn}
% \begin{syntax}
% \cs{int_do_while:nn} \Arg{integer relation} \Arg{code}
% \end{syntax}
% Places the \meta{code} in the input stream for \TeX{} to process, and
% then evaluates the \meta{integer relation}
% as described for \cs{int_compare:nTF}.
% If the test is \texttt{true} then the \meta{code} is inserted
% into the input stream again and a loop occurs until the
% \meta{relation} is \texttt{false}.
% \end{function}
%
% \begin{function}[updated = 2013-01-13, rEXP]{\int_until_do:nn}
% \begin{syntax}
% \cs{int_until_do:nn} \Arg{integer relation} \Arg{code}
% \end{syntax}
% Evaluates the \meta{integer relation}
% as described for \cs{int_compare:nTF}, and then places the
% \meta{code} in the input stream if the \meta{relation} is
% \texttt{false}. After the \meta{code} has been processed by \TeX{} the
% test is repeated, and a loop occurs until the test is
% \texttt{true}.
% \end{function}
%
% \begin{function}[updated = 2013-01-13, rEXP]{\int_while_do:nn}
% \begin{syntax}
% \cs{int_while_do:nn} \Arg{integer relation} \Arg{code}
% \end{syntax}
% Evaluates the \meta{integer relation}
% as described for \cs{int_compare:nTF}, and then places the
% \meta{code} in the input stream if the \meta{relation} is
% \texttt{true}. After the \meta{code} has been processed by \TeX{} the
% test is repeated, and a loop occurs until the test is
% \texttt{false}.
% \end{function}
%
% \section{Integer step functions}
%
% \begin{function}[added = 2012-06-04, updated = 2018-04-22, rEXP]
% {\int_step_function:nN, \int_step_function:nnN, \int_step_function:nnnN}
% \begin{syntax}
% \cs{int_step_function:nN} \Arg{final value} \meta{function}
% \cs{int_step_function:nnN} \Arg{initial value} \Arg{final value} \meta{function}
% \cs{int_step_function:nnnN} \Arg{initial value} \Arg{step} \Arg{final value} \meta{function}
% \end{syntax}
% This function first evaluates the \meta{initial value}, \meta{step}
% and \meta{final value}, all of which should be integer expressions.
% The \meta{function} is then placed in front of each \meta{value}
% from the \meta{initial value} to the \meta{final value} in turn
% (using \meta{step} between each \meta{value}). The \meta{step} must
% be non-zero. If the \meta{step} is positive, the loop stops when
% the \meta{value} becomes larger than the \meta{final value}. If the
% \meta{step} is negative, the loop stops when the \meta{value}
% becomes smaller than the \meta{final value}. The \meta{function}
% should absorb one numerical argument. For example
% \begin{verbatim}
% \cs_set:Npn \my_func:n #1 { [I~saw~#1] \quad }
% \int_step_function:nnnN { 1 } { 1 } { 5 } \my_func:n
% \end{verbatim}
% would print
% \begin{quote}
% [I saw 1] \quad
% [I saw 2] \quad
% [I saw 3] \quad
% [I saw 4] \quad
% [I saw 5] \quad
% \end{quote}
%
% The functions \cs{int_step_function:nN} and \cs{int_step_function:nnN}
% both use a fixed \meta{step} of $1$, and in the case of
% \cs{int_step_function:nN} the \meta{initial value} is also fixed as
% $1$. These functions are provided as simple short-cuts for code clarity.
% \end{function}
%
% \begin{function}[added = 2012-06-04, updated = 2018-04-22]
% {\int_step_inline:nn, \int_step_inline:nnn, \int_step_inline:nnnn}
% \begin{syntax}
% \cs{int_step_inline:nn} \Arg{final value} \Arg{code}
% \cs{int_step_inline:nnn} \Arg{initial value} \Arg{final value} \Arg{code}
% \cs{int_step_inline:nnnn} \Arg{initial value} \Arg{step} \Arg{final value} \Arg{code}
% \end{syntax}
% This function first evaluates the \meta{initial value}, \meta{step}
% and \meta{final value}, all of which should be integer expressions.
% Then for each \meta{value} from the \meta{initial value} to the
% \meta{final value} in turn (using \meta{step} between each
% \meta{value}), the \meta{code} is inserted into the input stream
% with |#1| replaced by the current \meta{value}. Thus the
% \meta{code} should define a function of one argument~(|#1|).
%
% The functions \cs{int_step_inline:nn} and \cs{int_step_inline:nnn}
% both use a fixed \meta{step} of $1$, and in the case of
% \cs{int_step_inline:nn} the \meta{initial value} is also fixed as
% $1$. These functions are provided as simple short-cuts for code clarity.
% \end{function}
%
% \begin{function}[added = 2012-06-04, updated = 2018-04-22]
% {\int_step_variable:nNn, \int_step_variable:nnNn, \int_step_variable:nnnNn}
% \begin{syntax}
% \cs{int_step_variable:nNn} \Arg{final value} \meta{tl~var} \Arg{code}
% \cs{int_step_variable:nnNn} \Arg{initial value} \Arg{final value} \meta{tl~var} \Arg{code}
% \cs{int_step_variable:nnnNn} \Arg{initial value} \Arg{step} \Arg{final value} \meta{tl~var} \Arg{code}
% \end{syntax}
% This function first evaluates the \meta{initial value}, \meta{step}
% and \meta{final value}, all of which should be integer expressions.
% Then for each \meta{value} from the \meta{initial value} to the
% \meta{final value} in turn (using \meta{step} between each
% \meta{value}), the \meta{code} is inserted into the input stream,
% with the \meta{tl~var} defined as the current \meta{value}. Thus
% the \meta{code} should make use of the \meta{tl~var}.
%
% The functions \cs{int_step_variable:nNn} and \cs{int_step_variable:nnNn}
% both use a fixed \meta{step} of $1$, and in the case of
% \cs{int_step_variable:nNn} the \meta{initial value} is also fixed as
% $1$. These functions are provided as simple short-cuts for code clarity.
% \end{function}
%
% \section{Formatting integers}
%
% Integers can be placed into the output stream with formatting. These
% conversions apply to any integer expressions.
%
% \begin{function}[updated = 2011-10-22, EXP]{\int_to_arabic:n}
% \begin{syntax}
% \cs{int_to_arabic:n} \Arg{integer expression}
% \end{syntax}
% Places the value of the \meta{integer expression} in the input
% stream as digits, with category code $12$ (other).
% \end{function}
%
% \begin{function}[updated = 2011-09-17, EXP]{\int_to_alph:n, \int_to_Alph:n}
% \begin{syntax}
% \cs{int_to_alph:n} \Arg{integer expression}
% \end{syntax}
% Evaluates the \meta{integer expression} and converts the result
% into a series of letters, which are then left in the input stream.
% The conversion rule uses the $26$ letters of the English
% alphabet, in order, adding letters when necessary to increase the total
% possible range of representable numbers. Thus
% \begin{verbatim}
% \int_to_alph:n { 1 }
% \end{verbatim}
% places |a| in the input stream,
% \begin{verbatim}
% \int_to_alph:n { 26 }
% \end{verbatim}
% is represented as |z| and
% \begin{verbatim}
% \int_to_alph:n { 27 }
% \end{verbatim}
% is converted to |aa|. For conversions using other alphabets, use
% \cs{int_to_symbols:nnn} to define an alphabet-specific
% function. The basic \cs{int_to_alph:n} and \cs{int_to_Alph:n}
% functions should not be modified.
% The resulting tokens are digits with category code $12$ (other) and
% letters with category code $11$ (letter).
% \end{function}
%
% \begin{function}[updated = 2011-09-17, EXP]{\int_to_symbols:nnn}
% \begin{syntax}
% \cs{int_to_symbols:nnn}
% ~~\Arg{integer expression} \Arg{total symbols}
% ~~\Arg{value to symbol mapping}
% \end{syntax}
% This is the low-level function for conversion of an
% \meta{integer expression} into a symbolic form (often
% letters). The \meta{total symbols} available should be given
% as an integer expression. Values are actually converted to symbols
% according to the \meta{value to symbol mapping}. This should be given
% as \meta{total symbols} pairs of entries, a number and the
% appropriate symbol. Thus the \cs{int_to_alph:n} function is defined
% as
% \begin{verbatim}
% \cs_new:Npn \int_to_alph:n #1
% {
% \int_to_symbols:nnn {#1} { 26 }
% {
% { 1 } { a }
% { 2 } { b }
% ...
% { 26 } { z }
% }
% }
% \end{verbatim}
% \end{function}
%
% \begin{function}[added = 2014-02-11, EXP]{\int_to_bin:n}
% \begin{syntax}
% \cs{int_to_bin:n} \Arg{integer expression}
% \end{syntax}
% Calculates the value of the \meta{integer expression} and places
% the binary representation of the result in the input stream.
% \end{function}
%
% \begin{function}[added = 2014-02-11, EXP]{\int_to_hex:n, \int_to_Hex:n}
% \begin{syntax}
% \cs{int_to_hex:n} \Arg{integer expression}
% \end{syntax}
% Calculates the value of the \meta{integer expression} and places
% the hexadecimal (base~$16$) representation of the result in the
% input stream. Letters are used for digits beyond~$9$: lower
% case letters for \cs{int_to_hex:n} and upper case ones for
% \cs{int_to_Hex:n}.
% The resulting tokens are digits with category code $12$ (other) and
% letters with category code $11$ (letter).
% \end{function}
%
% \begin{function}[added = 2014-02-11, EXP]{\int_to_oct:n}
% \begin{syntax}
% \cs{int_to_oct:n} \Arg{integer expression}
% \end{syntax}
% Calculates the value of the \meta{integer expression} and places
% the octal (base~$8$) representation of the result in the input
% stream.
% The resulting tokens are digits with category code $12$ (other) and
% letters with category code $11$ (letter).
% \end{function}
%
% \begin{function}[updated = 2014-02-11, EXP]
% {\int_to_base:nn, \int_to_Base:nn}
% \begin{syntax}
% \cs{int_to_base:nn} \Arg{integer expression} \Arg{base}
% \end{syntax}
% Calculates the value of the \meta{integer expression} and
% converts it into the appropriate representation in the \meta{base};
% the later may be given as an integer expression. For bases greater
% than $10$ the higher \enquote{digits} are represented by
% letters from the English alphabet: lower
% case letters for \cs{int_to_base:n} and upper case ones for
% \cs{int_to_Base:n}.
% The maximum \meta{base} value is $36$.
% The resulting tokens are digits with category code $12$ (other) and
% letters with category code $11$ (letter).
% \begin{texnote}
% This is a generic version of \cs{int_to_bin:n}, \emph{etc.}
% \end{texnote}
% \end{function}
%
% \begin{function}[updated = 2011-10-22, rEXP]{\int_to_roman:n, \int_to_Roman:n}
% \begin{syntax}
% \cs{int_to_roman:n} \Arg{integer expression}
% \end{syntax}
% Places the value of the \meta{integer expression} in the input
% stream as Roman numerals, either lower case (\cs{int_to_roman:n}) or
% upper case (\cs{int_to_Roman:n}). If the value is negative or zero,
% the output is empty. The Roman numerals are letters with category
% code $11$ (letter). The letters used are |mdclxvi|, repeated as
% needed: the notation with bars (such as $\bar{\mbox{v}}$ for $5000$)
% is \emph{not} used. For instance \cs{int_to_roman:n} |{| 8249 |}|
% expands to |mmmmmmmmccxlix|.
% \end{function}
%
% \section{Converting from other formats to integers}
%
% \begin{function}[updated = 2014-08-25, EXP]{\int_from_alph:n}
% \begin{syntax}
% \cs{int_from_alph:n} \Arg{letters}
% \end{syntax}
% Converts the \meta{letters} into the integer (base~$10$)
% representation and leaves this in the input stream. The
% \meta{letters} are first converted to a string, with no expansion.
% Lower and upper case letters from the English alphabet may be used,
% with \enquote{a} equal to $1$ through to \enquote{z} equal to $26$.
% The function also accepts a leading sign, made of |+| and~|-|. This
% is the inverse function of \cs{int_to_alph:n} and
% \cs{int_to_Alph:n}.
% \end{function}
%
% \begin{function}[added = 2014-02-11, updated = 2014-08-25, EXP]
% {\int_from_bin:n}
% \begin{syntax}
% \cs{int_from_bin:n} \Arg{binary number}
% \end{syntax}
% Converts the \meta{binary number} into the integer (base~$10$)
% representation and leaves this in the input stream.
% The \meta{binary number} is first converted to a string, with no
% expansion. The function accepts a leading sign, made of |+|
% and~|-|, followed by binary digits. This is the inverse function
% of \cs{int_to_bin:n}.
% \end{function}
%
% \begin{function}[added = 2014-02-11, updated = 2014-08-25, EXP]
% {\int_from_hex:n}
% \begin{syntax}
% \cs{int_from_hex:n} \Arg{hexadecimal number}
% \end{syntax}
% Converts the \meta{hexadecimal number} into the integer (base~$10$)
% representation and leaves this in the input stream. Digits greater
% than $9$ may be represented in the \meta{hexadecimal number} by
% upper or lower case letters. The \meta{hexadecimal number} is first
% converted to a string, with no expansion. The function also accepts
% a leading sign, made of |+| and~|-|. This is the inverse function
% of \cs{int_to_hex:n} and \cs{int_to_Hex:n}.
% \end{function}
%
% \begin{function}[added = 2014-02-11, updated = 2014-08-25, EXP]
% {\int_from_oct:n}
% \begin{syntax}
% \cs{int_from_oct:n} \Arg{octal number}
% \end{syntax}
% Converts the \meta{octal number} into the integer (base~$10$)
% representation and leaves this in the input stream.
% The \meta{octal number} is first converted to a string, with no
% expansion. The function accepts a leading sign, made of |+|
% and~|-|, followed by octal digits. This is the inverse function
% of \cs{int_to_oct:n}.
% \end{function}
%
% \begin{function}[updated = 2014-08-25, updated = 2014-08-25, EXP]
% {\int_from_roman:n}
% \begin{syntax}
% \cs{int_from_roman:n} \Arg{roman numeral}
% \end{syntax}
% Converts the \meta{roman numeral} into the integer (base~$10$)
% representation and leaves this in the input stream. The \meta{roman
% numeral} is first converted to a string, with no expansion. The
% \meta{roman numeral} may be in upper or lower case; if the numeral
% contains characters besides |mdclxvi| or |MDCLXVI| then the
% resulting value is $-1$. This is the inverse function of
% \cs{int_to_roman:n} and \cs{int_to_Roman:n}.
% \end{function}
%
% \begin{function}[updated = 2014-08-25, EXP]{\int_from_base:nn}
% \begin{syntax}
% \cs{int_from_base:nn} \Arg{number} \Arg{base}
% \end{syntax}
% Converts the \meta{number} expressed in \meta{base} into the
% appropriate value in base $10$. The \meta{number} is first
% converted to a string, with no expansion. The \meta{number} should
% consist of digits and letters (either lower or upper case), plus
% optionally a leading sign. The maximum \meta{base} value is $36$.
% This is the inverse function of \cs{int_to_base:nn} and
% \cs{int_to_Base:nn}.
% \end{function}
%
% \section{Random integers}
%
% \begin{function}[EXP, added = 2016-12-06, updated = 2018-04-27]{\int_rand:nn}
% \begin{syntax}
% \cs{int_rand:nn} \Arg{intexpr_1} \Arg{intexpr_2}
% \end{syntax}
% Evaluates the two \meta{integer expressions} and produces a
% pseudo-random number between the two (with bounds included).
% This is not available in older versions of \XeTeX{}.
% \end{function}
%
% \begin{function}[EXP, added = 2018-05-05]{\int_rand:n}
% \begin{syntax}
% \cs{int_rand:n} \Arg{intexpr}
% \end{syntax}
% Evaluates the \meta{integer expression} then produces a
% pseudo-random number between $1$ and the \meta{intexpr} (included).
% This is not available in older versions of \XeTeX{}.
% \end{function}
%
% \section{Viewing integers}
%
% \begin{function}{\int_show:N, \int_show:c}
% \begin{syntax}
% \cs{int_show:N} \meta{integer}
% \end{syntax}
% Displays the value of the \meta{integer} on the terminal.
% \end{function}
%
% \begin{function}[added = 2011-11-22, updated = 2015-08-07]{\int_show:n}
% \begin{syntax}
% \cs{int_show:n} \Arg{integer expression}
% \end{syntax}
% Displays the result of evaluating the \meta{integer expression}
% on the terminal.
% \end{function}
%
% \begin{function}[added = 2014-08-22, updated = 2015-08-03]{\int_log:N, \int_log:c}
% \begin{syntax}
% \cs{int_log:N} \meta{integer}
% \end{syntax}
% Writes the value of the \meta{integer} in the log file.
% \end{function}
%
% \begin{function}[added = 2014-08-22, updated = 2015-08-07]{\int_log:n}
% \begin{syntax}
% \cs{int_log:n} \Arg{integer expression}
% \end{syntax}
% Writes the result of evaluating the \meta{integer expression}
% in the log file.
% \end{function}
%
% \section{Constant integers}
%
% \begin{variable}[added = 2018-05-07]{\c_zero_int, \c_one_int}
% Integer values used with primitive tests and assignments: their
% self-terminating nature makes these more convenient and faster than
% literal numbers.
% \end{variable}
%
% \begin{variable}{\c_max_int}
% The maximum value that can be stored as an integer.
% \end{variable}
%
% \begin{variable}{\c_max_register_int}
% Maximum number of registers.
% \end{variable}
%
% \begin{variable}{\c_max_char_int}
% Maximum character code completely supported by the engine.
% \end{variable}
%
% \section{Scratch integers}
%
% \begin{variable}{\l_tmpa_int, \l_tmpb_int}
% Scratch integer for local assignment. These are never used by
% the kernel code, and so are safe for use with any \LaTeX3-defined
% function. However, they may be overwritten by other non-kernel
% code and so should only be used for short-term storage.
% \end{variable}
%
% \begin{variable}{\g_tmpa_int, \g_tmpb_int}
% Scratch integer for global assignment. These are never used by
% the kernel code, and so are safe for use with any \LaTeX3-defined
% function. However, they may be overwritten by other non-kernel
% code and so should only be used for short-term storage.
% \end{variable}
%
% \subsection{Direct number expansion}
%
% \begin{function}[EXP, added = 2018-03-27]{\int_value:w}
% \begin{syntax}
% \cs{int_value:w} \meta{integer}
% \cs{int_value:w} \meta{integer denotation} \meta{optional space}
% \end{syntax}
% Expands the following tokens until an \meta{integer} is formed, and
% leaves a normalized form (no leading sign except for negative
% numbers, no leading digit~|0| except for zero) in the input stream
% as category code $12$ (other) characters. The \meta{integer} can
% consist of any number of signs (with intervening spaces) followed
% by
% \begin{itemize}
% \item an integer variable (in fact, any \TeX{} register except
% \tn{toks}) or
% \item explicit digits (or by |'|\meta{octal digits} or |"|\meta{hexadecimal digits} or |`|\meta{character}).
% \end{itemize}
% In this last case expansion stops once a non-digit is found; if that is a
% space it is removed as in \texttt{f}-expansion, and so \cs{exp_stop_f:}
% may be employed as an end marker. Note that protected functions
% \emph{are} expanded by this process.
%
% This function requires exactly one expansion to produce a value, and so
% is suitable for use in cases where a number is required \enquote{directly}.
% In general, \cs{int_eval:n} is the preferred approach to generating
% numbers.
% \begin{texnote}
% This is the \TeX{} primitive \tn{number}.
% \end{texnote}
% \end{function}
%
% \section{Primitive conditionals}
%
% \begin{function}[EXP]{\if_int_compare:w}
% \begin{syntax}
% \cs{if_int_compare:w} \meta{integer_1} \meta{relation} \meta{integer_2}
% ~~\meta{true code}
% \cs{else:}
% ~~\meta{false code}
% \cs{fi:}
% \end{syntax}
% Compare two integers using \meta{relation}, which must be one of
% |=|, |<| or |>| with category code $12$.
% The \cs{else:} branch is optional.
% \begin{texnote}
% These are both names for the \TeX{} primitive \tn{ifnum}.
% \end{texnote}
% \end{function}
%
% \begin{function}[EXP]{\if_case:w, \or:}
% \begin{syntax}
% \cs{if_case:w} \meta{integer} \meta{case_0}
% ~~\cs{or:} \meta{case_1}
% ~~\cs{or:} |...|
% ~~\cs{else:} \meta{default}
% \cs{fi:}
% \end{syntax}
% Selects a case to execute based on the value of the \meta{integer}. The
% first case (\meta{case_0}) is executed if \meta{integer} is $0$, the second
% (\meta{case_1}) if the \meta{integer} is $1$, \emph{etc.} The
% \meta{integer} may be a literal, a constant or an integer
% expression (\emph{e.g.}~using \cs{int_eval:n}).
% \begin{texnote}
% These are the \TeX{} primitives \tn{ifcase} and \tn{or}.
% \end{texnote}
% \end{function}
%
% \begin{function}[EXP]{\if_int_odd:w}
% \begin{syntax}
% \cs{if_int_odd:w} \meta{tokens} \meta{optional space}
% ~~\meta{true code}
% \cs{else:}
% ~~\meta{true code}
% \cs{fi:}
% \end{syntax}
% Expands \meta{tokens} until a non-numeric token or a space is found, and
% tests whether the resulting \meta{integer} is odd. If so, \meta{true code}
% is executed. The \cs{else:} branch is optional.