algorithms.tex

%!TEX root = std.tex
\rSec0[algorithms]{Algorithms library}

\rSec1[algorithms.general]{General}

\pnum
This Clause describes components that \Cpp{} programs may use to perform
algorithmic operations on containers\iref{containers} and other sequences.

\pnum
The following subclauses describe components for
non-modifying sequence operations,
mutating sequence operations,
sorting and related operations,
and algorithms from the ISO C library,
as summarized in \tref{algorithms.summary}.

\begin{libsumtab}{Algorithms library summary}{algorithms.summary}
\ref{algorithms.requirements}  & Algorithms requirements           & \\
\ref{algorithms.parallel}      & Parallel algorithms               & \\ \rowsep
\ref{alg.nonmodifying}         & Non-modifying sequence operations & \tcode{<algorithm>} \\
\ref{alg.modifying.operations} & Mutating sequence operations      & \\
\ref{alg.sorting}              & Sorting and related operations    & \\ \rowsep
\ref{numeric.ops}              & Generalized numeric operations    & \tcode{<numeric>} \\ \rowsep
\ref{alg.c.library}            & C library algorithms              & \tcode{<cstdlib>} \\
\end{libsumtab}

\rSec1[algorithms.requirements]{Algorithms requirements}
\pnum
All of the algorithms
are separated from the particular implementations of data structures and
are parameterized by iterator types.
Because of this, they can work with program-defined data structures,
as long as these data structures have iterator types
satisfying the assumptions on the algorithms.

\pnum
The entities defined in the \tcode{std::ranges} namespace in this Clause
are not found by argument-dependent name lookup\iref{basic.lookup.argdep}.
When found by unqualified\iref{basic.lookup.unqual} name lookup
for the \grammarterm{postfix-expression} in a function call\iref{expr.call},
they inhibit argument-dependent name lookup.

\begin{example}
\begin{codeblock}
void foo() {
  using namespace std::ranges;
  std::vector<int> vec{1,2,3};
  find(begin(vec), end(vec), 2); // \#1
}
\end{codeblock}
The function call expression at \tcode{\#1} invokes \tcode{std::ranges::find},
not \tcode{std::find}, despite that
(a) the iterator type returned from \tcode{begin(vec)} and \tcode{end(vec)}
may be associated with namespace \tcode{std} and
(b) \tcode{std::find} is more specialized\iref{temp.func.order} than
\tcode{std::ranges::find} since the former requires
its first two parameters to have the same type.
\end{example}

\pnum
For purposes of determining the existence of data races,
algorithms shall not modify objects referenced through an iterator argument
unless the specification requires such modification.

\pnum
Throughout this Clause, where the template parameters are not constrained,
the names of template parameters are used to express type requirements.
\begin{itemize}
\item
  If an algorithm's template parameter is named
  \tcode{InputIterator},
  \tcode{InputIterator1}, or
  \tcode{Input\-Iterator2},
  the template argument shall satisfy the
  \oldconcept{InputIterator} requirements\iref{input.iterators}.
\item
  If an algorithm's template parameter is named
  \tcode{OutputIterator},
  \tcode{OutputIterator1}, or
  \tcode{Output\-Iterator2},
  the template argument shall satisfy the
  \oldconcept{OutputIterator} requirements\iref{output.iterators}.
\item
  If an algorithm's template parameter is named
  \tcode{ForwardIterator},
  \tcode{ForwardIterator1}, or
  \tcode{Forward\-Iterator2},
  the template argument shall satisfy the
  \oldconcept{ForwardIterator} requirements\iref{forward.iterators}.
\item
  If an algorithm's template parameter is named
  \tcode{BidirectionalIterator},
  \tcode{Bidirectional\-Iterator1}, or
  \tcode{BidirectionalIterator2},
  the template argument shall satisfy the
  \oldconcept{BidirectionalIterator} requirements\iref{bidirectional.iterators}.
\item
  If an algorithm's template parameter is named
  \tcode{RandomAccessIterator},
  \tcode{Random\-AccessIterator1}, or
  \tcode{RandomAccessIterator2},
  the template argument shall satisfy the
  \oldconcept{RandomAccessIterator} requirements\iref{random.access.iterators}.
\end{itemize}

\pnum
If an algorithm's \effects element specifies
that a value pointed to by any iterator passed as an argument is modified,
then that algorithm has an additional type requirement:
The type of that argument shall satisfy
the requirements of a mutable iterator\iref{iterator.requirements}.
\begin{note}
This requirement does not affect arguments that are named
\tcode{OutputIterator}, \tcode{OutputIterator1}, or \tcode{OutputIterator2},
because output iterators must always be mutable, nor
does it affect arguments that are constrained,
for which mutability requirements are expressed explicitly.
\end{note}

\pnum
Both in-place and copying versions are provided for certain algorithms.%
\footnote{The decision whether to include a copying version was
usually based on complexity considerations.
When the cost of doing the operation dominates the cost of copy,
the copying version is not included.
For example, \tcode{sort_copy} is not included
because the cost of sorting is much more significant,
and users might as well do \tcode{copy} followed by \tcode{sort}.}
When such a version is provided for \textit{algorithm} it is called
\textit{algorithm\tcode{_copy}}.
Algorithms that take predicates end with the suffix \tcode{_if}
(which follows the suffix \tcode{_copy}).

\pnum
When not otherwise constrained, the \tcode{Predicate} parameter is used
whenever an algorithm expects a function object\iref{function.objects} that,
when applied to the result of dereferencing the corresponding iterator,
returns a value testable as \tcode{true}.
In other words,
if an algorithm takes \tcode{Predicate pred} as its argument and
\tcode{first} as its iterator argument with value type \tcode{T},
it should work correctly in the construct
\tcode{pred(*first)} contextually converted to \tcode{bool}\iref{conv}.
The function object \tcode{pred} shall not apply any non-constant function
through the dereferenced iterator.
Given a glvalue \tcode{u} of type (possibly \tcode{const}) \tcode{T}
that designates the same object as \tcode{*first},
\tcode{pred(u)} shall be a valid expression
that is equal to \tcode{pred(*first)}.

\pnum
When not otherwise constrained, the \tcode{BinaryPredicate} parameter is used
whenever an algorithm expects a function object that when applied
to the result of dereferencing two corresponding iterators or
to dereferencing an iterator and type \tcode{T}
when \tcode{T} is part of the signature
returns a value testable as \tcode{true}.
In other words,
if an algorithm takes \tcode{BinaryPredicate binary_pred} as its argument and
\tcode{first1} and \tcode{first2} as its iterator arguments
with respective value types \tcode{T1} and \tcode{T2},
it should work correctly in the construct
\tcode{binary_pred(*first1, *first2)} contextually converted to \tcode{bool}\iref{conv}.
Unless otherwise specified,
\tcode{BinaryPredicate} always takes the first iterator's \tcode{value_type}
as its first argument, that is, in those cases when \tcode{T value}
is part of the signature, it should work correctly in the construct
\tcode{binary_pred(*first1, value)} contextually converted to \tcode{bool}\iref{conv}.
\tcode{binary_pred} shall not apply any non-constant function
through the dereferenced iterators.
Given a glvalue \tcode{u} of type (possibly \tcode{const}) \tcode{T1}
that designates the same object as \tcode{*first1}, and
a glvalue \tcode{v} of type (possibly \tcode{const}) \tcode{T2}
that designates the same object as \tcode{*first2},
\tcode{binary_pred(u, *first2)},
\tcode{binary_pred(*first1, v)}, and
\tcode{binary_pred(u, v)}
shall each be a valid expression that is equal to
\tcode{binary_pred(*first1, *first2)}, and
\tcode{binary_pred(u, value)}
shall be a valid expression that is equal to
\tcode{binary_pred(*first1, value)}.

\pnum
The parameters
\tcode{UnaryOperation},
\tcode{BinaryOperation},
\tcode{BinaryOperation1},
and \tcode{BinaryOperation2}
are used
whenever an algorithm expects a function object\iref{function.objects}.

\pnum
\begin{note}
Unless otherwise specified, algorithms that take function objects as arguments
are permitted to copy those function objects freely.
Programmers for whom object identity is important should consider
using a wrapper class that points to a noncopied implementation object
such as \tcode{reference_wrapper<T>}\iref{refwrap}, or some equivalent solution.
\end{note}

\pnum
When the description of an algorithm gives an expression such as
\tcode{*first == value} for a condition, the expression
shall evaluate to either \tcode{true} or \tcode{false} in boolean contexts.

\pnum
In the description of the algorithms, operator \tcode{+}
is used for some of the iterator categories
for which it does not have to be defined.
In these cases the semantics of \tcode{a + n} are the same as those of
\begin{codeblock}
auto tmp = a;
for (; n < 0; ++n) --tmp;
for (; n > 0; --n) ++tmp;
return tmp;
\end{codeblock}
Similarly, operator \tcode{-} is used
for some combinations of iterators and sentinel types
for which it does not have to be defined.
If \range{a}{b} denotes a range,
the semantics of \tcode{b - a} in these cases are the same as those of
\begin{codeblock}
iter_difference_t<remove_reference_t<decltype(a)>> n = 0;
for (auto tmp = a; tmp != b; ++tmp) ++n;
return n;
\end{codeblock}
and if \range{b}{a} denotes a range, the same as those of
\begin{codeblock}
iter_difference_t<remove_reference_t<decltype(b)>> n = 0;
for (auto tmp = b; tmp != a; ++tmp) --n;
return n;
\end{codeblock}

\pnum
In the description of algorithm return values,
a sentinel value \tcode{s} denoting the end of a range \range{i}{s}
is sometimes returned where an iterator is expected.
In these cases,
the semantics are as if the sentinel is converted into an iterator using
\tcode{ranges::next(i, s)}.

\pnum
Overloads of algorithms that take \libconcept{Range} arguments\iref{range.range}
behave as if they are implemented by calling \tcode{ranges::begin} and
\tcode{ranges::end} on the \libconcept{Range}(s) and
dispatching to the overload in namespace \tcode{ranges}
that takes separate iterator and sentinel arguments.

\pnum
The number and order of deducible template parameters for algorithm declarations
are unspecified, except where explicitly stated otherwise.
\begin{note}
Consequently, the algorithms may not
be called with explicitly-specified template argument lists.
\end{note}

\pnum
The class templates
\tcode{binary_transform_result},
\tcode{for_each_result},
\tcode{minmax_result},
\tcode{mismatch_result},
\tcode{copy_result}, and
\tcode{partition_copy_result}
have the template parameters, data members, and special members specified below.
They have no base classes or members other than those specified.

\rSec1[algorithms.parallel]{Parallel algorithms}

\pnum
This subclause describes components that \Cpp{} programs may use
to perform operations on containers and other sequences in parallel.

\rSec2[algorithms.parallel.defns]{Terms and definitions}
\pnum
A \defn{parallel algorithm} is a function template listed in this document
with a template parameter named \tcode{ExecutionPolicy}.

\pnum
Parallel algorithms access objects indirectly accessible via their arguments
by invoking the following functions:
\begin{itemize}
\item
  All operations of the categories of the iterators
  that the algorithm is instantiated with.
\item
  Operations on those sequence elements that are required by its specification.
\item
  User-provided function objects
  to be applied during the execution of the algorithm,
  if required by the specification.
\item
  Operations on those function objects required by the specification.
\begin{note}
See~\ref{algorithms.requirements}.
\end{note}
\end{itemize}
These functions are herein called \defn{element access functions}.
\begin{example}
The \tcode{sort} function may invoke the following element access functions:
\begin{itemize}
\item
  Operations of the random-access iterator of the actual template argument
  (as per \ref{random.access.iterators}),
  as implied by the name of the template parameter \tcode{RandomAccessIterator}.
\item
  The \tcode{swap} function on the elements of the sequence
  (as per the preconditions specified in \ref{sort}).
\item
  The user-provided \tcode{Compare} function object.
\end{itemize}
\end{example}

\pnum
A standard library function is \defn{vectorization-unsafe}
if it is specified to synchronize with another function invocation, or
another function invocation is specified to synchronize with it,
and if it is not a memory allocation or deallocation function.
\begin{note}
Implementations must ensure that internal synchronization
inside standard library functions does not prevent forward progress
when those functions are executed by threads of execution
with weakly parallel forward progress guarantees.
\end{note}
\begin{example}
\begin{codeblock}
int x = 0;
std::mutex m;
void f() {
  int a[] = {1,2};
  std::for_each(std::execution::par_unseq, std::begin(a), std::end(a), [&](int) {
    std::lock_guard<mutex> guard(m);            // incorrect: \tcode{lock_guard} constructor calls \tcode{m.lock()}
  ++x;
  });
}
\end{codeblock}
The above program may result in two consecutive calls to \tcode{m.lock()}
on the same thread of execution (which may deadlock),
because the applications of the function object are not guaranteed
to run on different threads of execution.
\end{example}

\rSec2[algorithms.parallel.user]{Requirements on user-provided function objects}

\pnum
Unless otherwise specified,
function objects passed into parallel algorithms as objects of type
\tcode{Predicate},
\tcode{BinaryPredicate},
\tcode{Compare},
\tcode{UnaryOperation},
\tcode{BinaryOperation},
\tcode{BinaryOperation1},
\tcode{BinaryOperation2}, and
the operators used by the analogous overloads to these parallel algorithms
that could be formed by the invocation
with the specified default predicate or operation (where applicable)
shall not directly or indirectly modify objects via their arguments,
nor shall they rely on the identity of the provided objects.

\rSec2[algorithms.parallel.exec]{Effect of execution policies on algorithm execution}

\pnum
Parallel algorithms have
template parameters named \tcode{ExecutionPolicy}\iref{execpol}
which describe the manner in which the execution of these algorithms may be
parallelized and the manner in which they apply the element access functions.

\pnum
If an object is modified by an element access function,
the algorithm will perform no other unsynchronized accesses to that object.
The modifying element access functions are those
which are specified as modifying the object.
\begin{note}
For example,
\tcode{swap()},
\tcode{++},
\tcode{--},
\tcode{@=}, and
assignments
modify the object.
For the assignment and \tcode{@=} operators, only the left argument is modified.
\end{note}

\pnum
Unless otherwise stated, implementations may make arbitrary copies of elements
(with type \tcode{T}) from sequences
where \tcode{is_trivially_copy_constructible_v<T>}
and \tcode{is_trivially_destructible_v<T>} are \tcode{true}.
\begin{note}
This implies that user-supplied function objects should not rely on
object identity of arguments for such input sequences.
Users for whom the object identity of the arguments to these function objects
is important should consider using a wrapping iterator
that returns a non-copied implementation object
such as \tcode{reference_wrapper<T>}\iref{refwrap}
or some equivalent solution.
\end{note}

\pnum
The invocations of element access functions in parallel algorithms invoked with
an execution policy object of type \tcode{execution::sequenced_policy} all occur
in the calling thread of execution.
\begin{note}
The invocations are not interleaved; see~\ref{intro.execution}.
\end{note}

\pnum
The invocations of element access functions in parallel algorithms invoked with
an execution policy object of type \tcode{execution::unsequenced_policy}
are permitted to execute in an unordered fashion
in the calling thread of execution,
unsequenced with respect to one another in the calling thread of execution.
\begin{note}
This means that multiple function object invocations
may be interleaved on a single thread of execution,
which overrides the usual guarantee from \ref{intro.execution}
that function executions do not overlap with one another.
\end{note}
The behavior of a program is undefined if
it invokes a vectorization-unsafe standard library function
from user code
called from a \tcode{execution::unsequenced_policy} algorithm.
\begin{note}
Because \tcode{execution::unsequenced_policy} allows
the execution of element access functions
to be interleaved on a single thread of execution,
blocking synchronization, including the use of mutexes, risks deadlock.
\end{note}

\pnum
The invocations of element access functions in parallel algorithms invoked with
an execution policy object of type \tcode{execution::parallel_policy}
are permitted to execute either
in the invoking thread of execution or
in a thread of execution implicitly created by the library
to support parallel algorithm execution.
If the threads of execution created by \tcode{thread}\iref{thread.thread.class}
provide concurrent forward progress guarantees\iref{intro.progress},
then a thread of execution implicitly created by the library will provide
parallel forward progress guarantees;
otherwise, the provided forward progress guarantee is
\impldef{forward progress guarantees for
implicit threads of parallel algorithms (if not defined for \tcode{thread})}.
Any such invocations executing in the same thread of execution
are indeterminately sequenced with respect to each other.
\begin{note}
It is the caller's responsibility to ensure
that the invocation does not introduce data races or deadlocks.
\end{note}
\begin{example}
\begin{codeblock}
int a[] = {0,1};
std::vector<int> v;
std::for_each(std::execution::par, std::begin(a), std::end(a), [&](int i) {
  v.push_back(i*2+1);                                 // incorrect: data race
});
\end{codeblock}
The program above has a data race because of the unsynchronized access to the
container \tcode{v}.
\end{example}
\begin{example}
\begin{codeblock}
std::atomic<int> x{0};
int a[] = {1,2};
std::for_each(std::execution::par, std::begin(a), std::end(a), [&](int) {
  x.fetch_add(1, std::memory_order::relaxed);
  // spin wait for another iteration to change the value of \tcode{x}
  while (x.load(std::memory_order::relaxed) == 1) { } // incorrect: assumes execution order
});
\end{codeblock}
The above example depends on the order of execution of the iterations, and
will not terminate if both iterations are executed sequentially
on the same thread of execution.
\end{example}
\begin{example}
\begin{codeblock}
int x = 0;
std::mutex m;
int a[] = {1,2};
std::for_each(std::execution::par, std::begin(a), std::end(a), [&](int) {
  std::lock_guard<mutex> guard(m);
  ++x;
});
\end{codeblock}
The above example synchronizes access to object \tcode{x}
ensuring that it is incremented correctly.
\end{example}

\pnum
The invocations of element access functions in parallel algorithms invoked with
an execution policy object of type \tcode{execution::parallel_unsequenced_policy} are
permitted to execute
in an unordered fashion in unspecified threads of execution, and
unsequenced with respect to one another within each thread of execution.
These threads of execution are
either the invoking thread of execution
or threads of execution implicitly created by the library;
the latter will provide weakly parallel forward progress guarantees.
\begin{note}
This means that multiple function object invocations may be interleaved
on a single thread of execution,
which overrides the usual guarantee from \ref{intro.execution}
that function executions do not overlap with one another.
\end{note}
The behavior of a program is undefined if
it invokes a vectorization-unsafe standard library function
from user code
called from a \tcode{execution::parallel_unsequenced_policy} algorithm.
\begin{note}
Because \tcode{execution::parallel_unsequenced_policy} allows
the execution of element access functions
to be interleaved on a single thread of execution,
blocking synchronization, including the use of mutexes, risks deadlock.
\end{note}

\pnum
\begin{note}
The semantics of invocation with
\tcode{execution::unsequenced_policy},
\tcode{execution::parallel_policy}, or
\tcode{execution::parallel_unsequenced_policy}
allow the implementation to fall back to sequential execution
if the system cannot parallelize an algorithm invocation,
e.g., due to lack of resources.
\end{note}

\pnum
If an invocation of a parallel algorithm uses threads of execution
implicitly created by the library,
then the invoking thread of execution will either
\begin{itemize}
\item
  temporarily block
  with forward progress guarantee delegation\iref{intro.progress}
  on the completion of these library-managed threads of execution, or
\item
  eventually execute an element access function;
\end{itemize}
the thread of execution will continue to do so until the algorithm is finished.
\begin{note}
In blocking with forward progress guarantee delegation in this context,
a thread of execution created by the library
is considered to have finished execution
as soon as it has finished the execution
of the particular element access function
that the invoking thread of execution logically depends on.
\end{note}

\pnum
The semantics of parallel algorithms invoked with an execution policy object of
\impldef{additional execution policies supported by parallel algorithms} type
are \impldef{semantics of parallel algorithms invoked with
imple\-men\-tation-defined execution policies}.

\rSec2[algorithms.parallel.exceptions]{Parallel algorithm exceptions}

\pnum
During the execution of a parallel algorithm,
if temporary memory resources are required for parallelization
and none are available, the algorithm throws a \tcode{bad_alloc} exception.

\pnum
During the execution of a parallel algorithm,
if the invocation of an element access function exits via an uncaught exception,
the behavior is determined by the \tcode{ExecutionPolicy}.

\rSec2[algorithms.parallel.overloads]{\tcode{ExecutionPolicy} algorithm overloads}

\pnum
Parallel algorithms are algorithm overloads.
Each parallel algorithm overload
has an additional template type parameter named \tcode{ExecutionPolicy},
which is the first template parameter.
Additionally, each parallel algorithm overload
has an additional function parameter of type \tcode{ExecutionPolicy\&\&},
which is the first function parameter.
\begin{note}
Not all algorithms have parallel algorithm overloads.
\end{note}

\pnum
Unless otherwise specified,
the semantics of \tcode{ExecutionPolicy} algorithm overloads
are identical to their overloads without.

\pnum
Unless otherwise specified,
the complexity requirements of \tcode{ExecutionPolicy} algorithm overloads
are relaxed from the complexity requirements of the overloads without
as follows:
when the guarantee says ``at most \placeholder{expr}'' or
``exactly \placeholder{expr}''
and does not specify the number of assignments or swaps, and
\placeholder{expr} is not already expressed with  \bigoh{} notation,
the complexity of the algorithm shall be \bigoh{\placeholder{expr}}.

\pnum
Parallel algorithms shall not participate in overload resolution unless
\tcode{is_execution_policy_v<remove_cvref_t<ExecutionPolicy>>} is \tcode{true}.

\rSec1[algorithm.syn]{Header \tcode{<algorithm>} synopsis}
\indexhdr{algorithm}%

\begin{codeblock}
#include <initializer_list>

namespace std {
  // \ref{alg.nonmodifying}, non-modifying sequence operations
  // \ref{alg.all.of}, all of
  template<class InputIterator, class Predicate>
    constexpr bool all_of(InputIterator first, InputIterator last, Predicate pred);
  template<class ExecutionPolicy, class ForwardIterator, class Predicate>
    bool all_of(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                ForwardIterator first, ForwardIterator last, Predicate pred);

  namespace ranges {
    template<InputIterator I, Sentinel<I> S, class Proj = identity,
             IndirectUnaryPredicate<projected<I, Proj>> Pred>
      constexpr bool all_of(I first, S last, Pred pred, Proj proj = {});
    template<InputRange R, class Proj = identity,
             IndirectUnaryPredicate<projected<iterator_t<R>, Proj>> Pred>
      constexpr bool all_of(R&& r, Pred pred, Proj proj = {});
  }

  // \ref{alg.any.of}, any of
  template<class InputIterator, class Predicate>
    constexpr bool any_of(InputIterator first, InputIterator last, Predicate pred);
  template<class ExecutionPolicy, class ForwardIterator, class Predicate>
    bool any_of(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                ForwardIterator first, ForwardIterator last, Predicate pred);

  namespace ranges {
    template<InputIterator I, Sentinel<I> S, class Proj = identity,
             IndirectUnaryPredicate<projected<I, Proj>> Pred>
      constexpr bool any_of(I first, S last, Pred pred, Proj proj = {});
    template<InputRange R, class Proj = identity,
             IndirectUnaryPredicate<projected<iterator_t<R>, Proj>> Pred>
      constexpr bool any_of(R&& r, Pred pred, Proj proj = {});
  }

  // \ref{alg.none.of}, none of
  template<class InputIterator, class Predicate>
    constexpr bool none_of(InputIterator first, InputIterator last, Predicate pred);
  template<class ExecutionPolicy, class ForwardIterator, class Predicate>
    bool none_of(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                 ForwardIterator first, ForwardIterator last, Predicate pred);

  namespace ranges {
    template<InputIterator I, Sentinel<I> S, class Proj = identity,
             IndirectUnaryPredicate<projected<I, Proj>> Pred>
      constexpr bool none_of(I first, S last, Pred pred, Proj proj = {});
    template<InputRange R, class Proj = identity,
             IndirectUnaryPredicate<projected<iterator_t<R>, Proj>> Pred>
      constexpr bool none_of(R&& r, Pred pred, Proj proj = {});
  }

  // \ref{alg.foreach}, for each
  template<class InputIterator, class Function>
    constexpr Function for_each(InputIterator first, InputIterator last, Function f);
  template<class ExecutionPolicy, class ForwardIterator, class Function>
    void for_each(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                  ForwardIterator first, ForwardIterator last, Function f);

  namespace ranges {
    template<class I, class F>
    struct for_each_result {
      [[no_unique_address]] I in;
      [[no_unique_address]] F fun;

      template<class I2, class F2>
        requires ConvertibleTo<const I&, I2> && ConvertibleTo<const F&, F2>
        operator for_each_result<I2, F2>() const & {
          return {in, fun};
        }

      template<class I2, class F2>
        requires ConvertibleTo<I, I2> && ConvertibleTo<F, F2>
        operator for_each_result<I2, F2>() && {
          return {std::move(in), std::move(fun)};
        }
    };

    template<InputIterator I, Sentinel<I> S, class Proj = identity,
             IndirectUnaryInvocable<projected<I, Proj>> Fun>
      constexpr for_each_result<I, Fun>
        for_each(I first, S last, Fun f, Proj proj = {});
    template<InputRange R, class Proj = identity,
             IndirectUnaryInvocable<projected<iterator_t<R>, Proj>> Fun>
      constexpr for_each_result<safe_iterator_t<R>, Fun>
        for_each(R&& r, Fun f, Proj proj = {});
  }

  template<class InputIterator, class Size, class Function>
    constexpr InputIterator for_each_n(InputIterator first, Size n, Function f);
  template<class ExecutionPolicy, class ForwardIterator, class Size, class Function>
    ForwardIterator for_each_n(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                               ForwardIterator first, Size n, Function f);

  // \ref{alg.find}, find
  template<class InputIterator, class T>
    constexpr InputIterator find(InputIterator first, InputIterator last,
                                 const T& value);
  template<class ExecutionPolicy, class ForwardIterator, class T>
    ForwardIterator find(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                         ForwardIterator first, ForwardIterator last,
                         const T& value);
  template<class InputIterator, class Predicate>
    constexpr InputIterator find_if(InputIterator first, InputIterator last,
                                    Predicate pred);
  template<class ExecutionPolicy, class ForwardIterator, class Predicate>
    ForwardIterator find_if(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                            ForwardIterator first, ForwardIterator last,
                            Predicate pred);
  template<class InputIterator, class Predicate>
    constexpr InputIterator find_if_not(InputIterator first, InputIterator last,
                                        Predicate pred);
  template<class ExecutionPolicy, class ForwardIterator, class Predicate>
    ForwardIterator find_if_not(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                                ForwardIterator first, ForwardIterator last,
                                Predicate pred);

  namespace ranges {
    template<InputIterator I, Sentinel<I> S, class T, class Proj = identity>
      requires IndirectRelation<ranges::equal_to, projected<I, Proj>, const T*>
      constexpr I find(I first, S last, const T& value, Proj proj = {});
    template<InputRange R, class T, class Proj = identity>
      requires IndirectRelation<ranges::equal_to, projected<iterator_t<R>, Proj>, const T*>
      constexpr safe_iterator_t<R>
        find(R&& r, const T& value, Proj proj = {});
    template<InputIterator I, Sentinel<I> S, class Proj = identity,
             IndirectUnaryPredicate<projected<I, Proj>> Pred>
      constexpr I find_if(I first, S last, Pred pred, Proj proj = {});
    template<InputRange R, class Proj = identity,
             IndirectUnaryPredicate<projected<iterator_t<R>, Proj>> Pred>
      constexpr safe_iterator_t<R>
        find_if(R&& r, Pred pred, Proj proj = {});
    template<InputIterator I, Sentinel<I> S, class Proj = identity,
             IndirectUnaryPredicate<projected<I, Proj>> Pred>
      constexpr I find_if_not(I first, S last, Pred pred, Proj proj = {});
    template<InputRange R, class Proj = identity,
             IndirectUnaryPredicate<projected<iterator_t<R>, Proj>> Pred>
      constexpr safe_iterator_t<R>
        find_if_not(R&& r, Pred pred, Proj proj = {});
  }

  // \ref{alg.find.end}, find end
  template<class ForwardIterator1, class ForwardIterator2>
    constexpr ForwardIterator1
      find_end(ForwardIterator1 first1, ForwardIterator1 last1,
               ForwardIterator2 first2, ForwardIterator2 last2);
  template<class ForwardIterator1, class ForwardIterator2, class BinaryPredicate>
    constexpr ForwardIterator1
      find_end(ForwardIterator1 first1, ForwardIterator1 last1,
               ForwardIterator2 first2, ForwardIterator2 last2,
               BinaryPredicate pred);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2>
    ForwardIterator1
      find_end(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
               ForwardIterator1 first1, ForwardIterator1 last1,
               ForwardIterator2 first2, ForwardIterator2 last2);
  template<class ExecutionPolicy, class ForwardIterator1,
           class ForwardIterator2, class BinaryPredicate>
    ForwardIterator1
      find_end(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
               ForwardIterator1 first1, ForwardIterator1 last1,
               ForwardIterator2 first2, ForwardIterator2 last2,
               BinaryPredicate pred);

  namespace ranges {
    template<ForwardIterator I1, Sentinel<I1> S1, ForwardIterator I2, Sentinel<I2> S2,
             class Pred = ranges::equal_to, class Proj1 = identity, class Proj2 = identity>
      requires IndirectlyComparable<I1, I2, Pred, Proj1, Proj2>
      constexpr subrange<I1>
        find_end(I1 first1, S1 last1, I2 first2, S2 last2, Pred pred = {},
                 Proj1 proj1 = {}, Proj2 proj2 = {});
    template<ForwardRange R1, ForwardRange R2,
             class Pred = ranges::equal_to, class Proj1 = identity, class Proj2 = identity>
      requires IndirectlyComparable<iterator_t<R1>, iterator_t<R2>, Pred, Proj1, Proj2>
      constexpr safe_subrange_t<R1>
        find_end(R1&& r1, R2&& r2, Pred pred = {},
                 Proj1 proj1 = {}, Proj2 proj2 = {});
  }

  // \ref{alg.find.first.of}, find first
  template<class InputIterator, class ForwardIterator>
    constexpr InputIterator
      find_first_of(InputIterator first1, InputIterator last1,
                    ForwardIterator first2, ForwardIterator last2);
  template<class InputIterator, class ForwardIterator, class BinaryPredicate>
    constexpr InputIterator
      find_first_of(InputIterator first1, InputIterator last1,
                    ForwardIterator first2, ForwardIterator last2,
                    BinaryPredicate pred);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2>
    ForwardIterator1
      find_first_of(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                    ForwardIterator1 first1, ForwardIterator1 last1,
                    ForwardIterator2 first2, ForwardIterator2 last2);
  template<class ExecutionPolicy, class ForwardIterator1,
           class ForwardIterator2, class BinaryPredicate>
    ForwardIterator1
      find_first_of(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                    ForwardIterator1 first1, ForwardIterator1 last1,
                    ForwardIterator2 first2, ForwardIterator2 last2,
                    BinaryPredicate pred);

  namespace ranges {
    template<InputIterator I1, Sentinel<I1> S1, ForwardIterator I2, Sentinel<I2> S2,
             class Proj1 = identity, class Proj2 = identity,
             IndirectRelation<projected<I1, Proj1>,
                              projected<I2, Proj2>> Pred = ranges::equal_to>
      constexpr I1 find_first_of(I1 first1, S1 last1, I2 first2, S2 last2,
                                 Pred pred = {},
                                 Proj1 proj1 = {}, Proj2 proj2 = {});
    template<InputRange R1, ForwardRange R2, class Proj1 = identity, class Proj2 = identity,
             IndirectRelation<projected<iterator_t<R1>, Proj1>,
                              projected<iterator_t<R2>, Proj2>> Pred = ranges::equal_to>
      constexpr safe_iterator_t<R1>
        find_first_of(R1&& r1, R2&& r2,
                      Pred pred = {},
                      Proj1 proj1 = {}, Proj2 proj2 = {});
  }

  // \ref{alg.adjacent.find}, adjacent find
  template<class ForwardIterator>
    constexpr ForwardIterator
      adjacent_find(ForwardIterator first, ForwardIterator last);
  template<class ForwardIterator, class BinaryPredicate>
    constexpr ForwardIterator
      adjacent_find(ForwardIterator first, ForwardIterator last,
                    BinaryPredicate pred);
  template<class ExecutionPolicy, class ForwardIterator>
    ForwardIterator
      adjacent_find(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                    ForwardIterator first, ForwardIterator last);
  template<class ExecutionPolicy, class ForwardIterator, class BinaryPredicate>
    ForwardIterator
      adjacent_find(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                    ForwardIterator first, ForwardIterator last,
                    BinaryPredicate pred);

  namespace ranges {
    template<ForwardIterator I, Sentinel<I> S, class Proj = identity,
             IndirectRelation<projected<I, Proj>> Pred = ranges::equal_to>
      constexpr I adjacent_find(I first, S last, Pred pred = {},
                                Proj proj = {});
    template<ForwardRange R, class Proj = identity,
             IndirectRelation<projected<iterator_t<R>, Proj>> Pred = ranges::equal_to>
      constexpr safe_iterator_t<R>
        adjacent_find(R&& r, Pred pred = {}, Proj proj = {});
  }

  // \ref{alg.count}, count
  template<class InputIterator, class T>
    constexpr typename iterator_traits<InputIterator>::difference_type
      count(InputIterator first, InputIterator last, const T& value);
  template<class ExecutionPolicy, class ForwardIterator, class T>
    typename iterator_traits<ForwardIterator>::difference_type
      count(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
            ForwardIterator first, ForwardIterator last, const T& value);
  template<class InputIterator, class Predicate>
    constexpr typename iterator_traits<InputIterator>::difference_type
      count_if(InputIterator first, InputIterator last, Predicate pred);
  template<class ExecutionPolicy, class ForwardIterator, class Predicate>
    typename iterator_traits<ForwardIterator>::difference_type
      count_if(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
               ForwardIterator first, ForwardIterator last, Predicate pred);

  namespace ranges {
    template<InputIterator I, Sentinel<I> S, class T, class Proj = identity>
      requires IndirectRelation<ranges::equal_to, projected<I, Proj>, const T*>
      constexpr iter_difference_t<I>
        count(I first, S last, const T& value, Proj proj = {});
    template<InputRange R, class T, class Proj = identity>
      requires IndirectRelation<ranges::equal_to, projected<iterator_t<R>, Proj>, const T*>
      constexpr iter_difference_t<iterator_t<R>>
        count(R&& r, const T& value, Proj proj = {});
    template<InputIterator I, Sentinel<I> S, class Proj = identity,
             IndirectUnaryPredicate<projected<I, Proj>> Pred>
      constexpr iter_difference_t<I>
        count_if(I first, S last, Pred pred, Proj proj = {});
    template<InputRange R, class Proj = identity,
             IndirectUnaryPredicate<projected<iterator_t<R>, Proj>> Pred>
      constexpr iter_difference_t<iterator_t<R>>
        count_if(R&& r, Pred pred, Proj proj = {});
  }

  // \ref{mismatch}, mismatch
  template<class InputIterator1, class InputIterator2>
    constexpr pair<InputIterator1, InputIterator2>
      mismatch(InputIterator1 first1, InputIterator1 last1,
               InputIterator2 first2);
  template<class InputIterator1, class InputIterator2, class BinaryPredicate>
    constexpr pair<InputIterator1, InputIterator2>
      mismatch(InputIterator1 first1, InputIterator1 last1,
               InputIterator2 first2, BinaryPredicate pred);
  template<class InputIterator1, class InputIterator2>
    constexpr pair<InputIterator1, InputIterator2>
      mismatch(InputIterator1 first1, InputIterator1 last1,
               InputIterator2 first2, InputIterator2 last2);
  template<class InputIterator1, class InputIterator2, class BinaryPredicate>
    constexpr pair<InputIterator1, InputIterator2>
      mismatch(InputIterator1 first1, InputIterator1 last1,
               InputIterator2 first2, InputIterator2 last2,
               BinaryPredicate pred);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2>
    pair<ForwardIterator1, ForwardIterator2>
      mismatch(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
               ForwardIterator1 first1, ForwardIterator1 last1,
               ForwardIterator2 first2);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
           class BinaryPredicate>
    pair<ForwardIterator1, ForwardIterator2>
      mismatch(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
               ForwardIterator1 first1, ForwardIterator1 last1,
               ForwardIterator2 first2, BinaryPredicate pred);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2>
    pair<ForwardIterator1, ForwardIterator2>
      mismatch(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
               ForwardIterator1 first1, ForwardIterator1 last1,
               ForwardIterator2 first2, ForwardIterator2 last2);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
           class BinaryPredicate>
    pair<ForwardIterator1, ForwardIterator2>
      mismatch(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
               ForwardIterator1 first1, ForwardIterator1 last1,
               ForwardIterator2 first2, ForwardIterator2 last2,
               BinaryPredicate pred);

  namespace ranges {
    template<class I1, class I2>
    struct mismatch_result {
      [[no_unique_address]] I1 in1;
      [[no_unique_address]] I2 in2;

      template<class II1, class II2>
        requires ConvertibleTo<const I1&, II1> && ConvertibleTo<const I2&, II2>
        operator mismatch_result<II1, II2>() const & {
          return {in1, in2};
        }

      template<class II1, class II2>
        requires ConvertibleTo<I1, II1> && ConvertibleTo<I2, II2>
        operator mismatch_result<II1, II2>() && {
          return {std::move(in1), std::move(in2)};
        }
    };

    template<InputIterator I1, Sentinel<I1> S1, InputIterator I2, Sentinel<I2> S2,
             class Proj1 = identity, class Proj2 = identity,
             IndirectRelation<projected<I1, Proj1>,
                              projected<I2, Proj2>> Pred = ranges::equal_to>
      constexpr mismatch_result<I1, I2>
        mismatch(I1 first1, S1 last1, I2 first2, S2 last2, Pred pred = {},
                 Proj1 proj1 = {}, Proj2 proj2 = {});
    template<InputRange R1, InputRange R2,
             class Proj1 = identity, class Proj2 = identity,
             IndirectRelation<projected<iterator_t<R1>, Proj1>,
                              projected<iterator_t<R2>, Proj2>> Pred = ranges::equal_to>
      constexpr mismatch_result<safe_iterator_t<R1>, safe_iterator_t<R2>>
        mismatch(R1&& r1, R2&& r2, Pred pred = {},
                 Proj1 proj1 = {}, Proj2 proj2 = {});
  }

  // \ref{alg.equal}, equal
  template<class InputIterator1, class InputIterator2>
    constexpr bool equal(InputIterator1 first1, InputIterator1 last1,
                         InputIterator2 first2);
  template<class InputIterator1, class InputIterator2, class BinaryPredicate>
    constexpr bool equal(InputIterator1 first1, InputIterator1 last1,
                         InputIterator2 first2, BinaryPredicate pred);
  template<class InputIterator1, class InputIterator2>
    constexpr bool equal(InputIterator1 first1, InputIterator1 last1,
                         InputIterator2 first2, InputIterator2 last2);
  template<class InputIterator1, class InputIterator2, class BinaryPredicate>
    constexpr bool equal(InputIterator1 first1, InputIterator1 last1,
                         InputIterator2 first2, InputIterator2 last2,
                         BinaryPredicate pred);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2>
    bool equal(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
               ForwardIterator1 first1, ForwardIterator1 last1,
               ForwardIterator2 first2);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
           class BinaryPredicate>
    bool equal(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
               ForwardIterator1 first1, ForwardIterator1 last1,
               ForwardIterator2 first2, BinaryPredicate pred);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2>
    bool equal(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
               ForwardIterator1 first1, ForwardIterator1 last1,
               ForwardIterator2 first2, ForwardIterator2 last2);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
           class BinaryPredicate>
    bool equal(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
               ForwardIterator1 first1, ForwardIterator1 last1,
               ForwardIterator2 first2, ForwardIterator2 last2,
               BinaryPredicate pred);

  namespace ranges {
    template<InputIterator I1, Sentinel<I1> S1, InputIterator I2, Sentinel<I2> S2,
             class Pred = ranges::equal_to, class Proj1 = identity, class Proj2 = identity>
      requires IndirectlyComparable<I1, I2, Pred, Proj1, Proj2>
      constexpr bool equal(I1 first1, S1 last1, I2 first2, S2 last2,
                           Pred pred = {},
                           Proj1 proj1 = {}, Proj2 proj2 = {});
    template<InputRange R1, InputRange R2, class Pred = ranges::equal_to,
             class Proj1 = identity, class Proj2 = identity>
      requires IndirectlyComparable<iterator_t<R1>, iterator_t<R2>, Pred, Proj1, Proj2>
      constexpr bool equal(R1&& r1, R2&& r2, Pred pred = {},
                           Proj1 proj1 = {}, Proj2 proj2 = {});
  }

  // \ref{alg.is.permutation}, is permutation
  template<class ForwardIterator1, class ForwardIterator2>
    constexpr bool is_permutation(ForwardIterator1 first1, ForwardIterator1 last1,
                                  ForwardIterator2 first2);
  template<class ForwardIterator1, class ForwardIterator2, class BinaryPredicate>
    constexpr bool is_permutation(ForwardIterator1 first1, ForwardIterator1 last1,
                                  ForwardIterator2 first2, BinaryPredicate pred);
  template<class ForwardIterator1, class ForwardIterator2>
    constexpr bool is_permutation(ForwardIterator1 first1, ForwardIterator1 last1,
                                  ForwardIterator2 first2, ForwardIterator2 last2);
  template<class ForwardIterator1, class ForwardIterator2, class BinaryPredicate>
    constexpr bool is_permutation(ForwardIterator1 first1, ForwardIterator1 last1,
                                  ForwardIterator2 first2, ForwardIterator2 last2,
                                  BinaryPredicate pred);

  namespace ranges {
    template<ForwardIterator I1, Sentinel<I1> S1, ForwardIterator I2,
             Sentinel<I2> S2, class Pred = ranges::equal_to, class Proj1 = identity,
             class Proj2 = identity>
      requires IndirectlyComparable<I1, I2, Pred, Proj1, Proj2>
      constexpr bool is_permutation(I1 first1, S1 last1, I2 first2, S2 last2,
                                    Pred pred = {},
                                    Proj1 proj1 = {}, Proj2 proj2 = {});
    template<ForwardRange R1, ForwardRange R2, class Pred = ranges::equal_to,
             class Proj1 = identity, class Proj2 = identity>
      requires IndirectlyComparable<iterator_t<R1>, iterator_t<R2>, Pred, Proj1, Proj2>
      constexpr bool is_permutation(R1&& r1, R2&& r2, Pred pred = {},
                                    Proj1 proj1 = {}, Proj2 proj2 = {});
  }

  // \ref{alg.search}, search
  template<class ForwardIterator1, class ForwardIterator2>
    constexpr ForwardIterator1
      search(ForwardIterator1 first1, ForwardIterator1 last1,
             ForwardIterator2 first2, ForwardIterator2 last2);
  template<class ForwardIterator1, class ForwardIterator2, class BinaryPredicate>
    constexpr ForwardIterator1
      search(ForwardIterator1 first1, ForwardIterator1 last1,
             ForwardIterator2 first2, ForwardIterator2 last2,
             BinaryPredicate pred);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2>
    ForwardIterator1
      search(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
             ForwardIterator1 first1, ForwardIterator1 last1,
             ForwardIterator2 first2, ForwardIterator2 last2);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
           class BinaryPredicate>
    ForwardIterator1
      search(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
             ForwardIterator1 first1, ForwardIterator1 last1,
             ForwardIterator2 first2, ForwardIterator2 last2,
             BinaryPredicate pred);

  namespace ranges {
    template<ForwardIterator I1, Sentinel<I1> S1, ForwardIterator I2,
             Sentinel<I2> S2, class Pred = ranges::equal_to,
             class Proj1 = identity, class Proj2 = identity>
      requires IndirectlyComparable<I1, I2, Pred, Proj1, Proj2>
      constexpr subrange<I1>
        search(I1 first1, S1 last1, I2 first2, S2 last2, Pred pred = {},
               Proj1 proj1 = {}, Proj2 proj2 = {});
    template<ForwardRange R1, ForwardRange R2, class Pred = ranges::equal_to,
             class Proj1 = identity, class Proj2 = identity>
      requires IndirectlyComparable<iterator_t<R1>, iterator_t<R2>, Pred, Proj1, Proj2>
      constexpr safe_subrange_t<R1>
        search(R1&& r1, R2&& r2, Pred pred = {},
               Proj1 proj1 = {}, Proj2 proj2 = {});
  }

  template<class ForwardIterator, class Size, class T>
    constexpr ForwardIterator
      search_n(ForwardIterator first, ForwardIterator last,
               Size count, const T& value);
  template<class ForwardIterator, class Size, class T, class BinaryPredicate>
    constexpr ForwardIterator
      search_n(ForwardIterator first, ForwardIterator last,
               Size count, const T& value,
               BinaryPredicate pred);
  template<class ExecutionPolicy, class ForwardIterator, class Size, class T>
    ForwardIterator
      search_n(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
               ForwardIterator first, ForwardIterator last,
               Size count, const T& value);
  template<class ExecutionPolicy, class ForwardIterator, class Size, class T,
           class BinaryPredicate>
    ForwardIterator
      search_n(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
               ForwardIterator first, ForwardIterator last,
               Size count, const T& value,
               BinaryPredicate pred);

  namespace ranges {
    template<ForwardIterator I, Sentinel<I> S, class T,
             class Pred = ranges::equal_to, class Proj = identity>
      requires IndirectlyComparable<I, const T*, Pred, Proj>
      constexpr subrange<I>
        search_n(I first, S last, iter_difference_t<I> count,
                 const T& value, Pred pred = {}, Proj proj = {});
    template<ForwardRange R, class T, class Pred = ranges::equal_to,
             class Proj = identity>
      requires IndirectlyComparable<iterator_t<R>, const T*, Pred, Proj>
      constexpr safe_subrange_t<R>
        search_n(R&& r, iter_difference_t<iterator_t<R>> count,
                 const T& value, Pred pred = {}, Proj proj = {});
  }

  template<class ForwardIterator, class Searcher>
    constexpr ForwardIterator
      search(ForwardIterator first, ForwardIterator last, const Searcher& searcher);

  // \ref{alg.modifying.operations}, mutating sequence operations
  // \ref{alg.copy}, copy
  template<class InputIterator, class OutputIterator>
    constexpr OutputIterator copy(InputIterator first, InputIterator last,
                                  OutputIterator result);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2>
    ForwardIterator2 copy(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                          ForwardIterator1 first, ForwardIterator1 last,
                          ForwardIterator2 result);

  namespace ranges {
    template<class I, class O>
    struct copy_result {
      [[no_unique_address]] I in;
      [[no_unique_address]] O out;

      template<class I2, class O2>
        requires ConvertibleTo<const I&, I2> && ConvertibleTo<const O&, O2>
        operator copy_result<I2, O2>() const & {
          return {in, out};
        }

      template<class I2, class O2>
        requires ConvertibleTo<I, I2> && ConvertibleTo<O, O2>
        operator copy_result<I2, O2>() && {
          return {std::move(in), std::move(out)};
        }
    };

    template<InputIterator I, Sentinel<I> S, WeaklyIncrementable O>
      requires IndirectlyCopyable<I, O>
      constexpr copy_result<I, O>
        copy(I first, S last, O result);
    template<InputRange R, WeaklyIncrementable O>
      requires IndirectlyCopyable<iterator_t<R>, O>
      constexpr copy_result<safe_iterator_t<R>, O>
        copy(R&& r, O result);
  }

  template<class InputIterator, class Size, class OutputIterator>
    constexpr OutputIterator copy_n(InputIterator first, Size n,
                                    OutputIterator result);
  template<class ExecutionPolicy, class ForwardIterator1, class Size,
           class ForwardIterator2>
    ForwardIterator2 copy_n(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                            ForwardIterator1 first, Size n,
                            ForwardIterator2 result);

  namespace ranges {
    template<class I, class O>
    using copy_n_result = copy_result<I, O>;

    template<InputIterator I, WeaklyIncrementable O>
      requires IndirectlyCopyable<I, O>
      constexpr copy_n_result<I, O>
        copy_n(I first, iter_difference_t<I> n, O result);
  }

  template<class InputIterator, class OutputIterator, class Predicate>
    constexpr OutputIterator copy_if(InputIterator first, InputIterator last,
                                     OutputIterator result, Predicate pred);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
           class Predicate>
    ForwardIterator2 copy_if(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                             ForwardIterator1 first, ForwardIterator1 last,
                             ForwardIterator2 result, Predicate pred);

  namespace ranges {
    template<class I, class O>
    using copy_if_result = copy_result<I, O>;

    template<InputIterator I, Sentinel<I> S, WeaklyIncrementable O, class Proj = identity,
             IndirectUnaryPredicate<projected<I, Proj>> Pred>
      requires IndirectlyCopyable<I, O>
      constexpr copy_if_result<I, O>
        copy_if(I first, S last, O result, Pred pred, Proj proj = {});
    template<InputRange R, WeaklyIncrementable O, class Proj = identity,
             IndirectUnaryPredicate<projected<iterator_t<R>, Proj>> Pred>
      requires IndirectlyCopyable<iterator_t<R>, O>
      constexpr copy_if_result<safe_iterator_t<R>, O>
        copy_if(R&& r, O result, Pred pred, Proj proj = {});
  }

  template<class BidirectionalIterator1, class BidirectionalIterator2>
    constexpr BidirectionalIterator2
      copy_backward(BidirectionalIterator1 first, BidirectionalIterator1 last,
                    BidirectionalIterator2 result);

  namespace ranges {
    template<class I1, class I2>
    using copy_backward_result = copy_result<I1, I2>;

    template<BidirectionalIterator I1, Sentinel<I1> S1, BidirectionalIterator I2>
      requires IndirectlyCopyable<I1, I2>
      constexpr copy_backward_result<I1, I2>
        copy_backward(I1 first, S1 last, I2 result);
    template<BidirectionalRange R, BidirectionalIterator I>
      requires IndirectlyCopyable<iterator_t<R>, I>
      constexpr copy_backward_result<safe_iterator_t<R>, I>
        copy_backward(R&& r, I result);
  }

  // \ref{alg.move}, move
  template<class InputIterator, class OutputIterator>
    constexpr OutputIterator move(InputIterator first, InputIterator last,
                                  OutputIterator result);
  template<class ExecutionPolicy, class ForwardIterator1,
           class ForwardIterator2>
    ForwardIterator2 move(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                          ForwardIterator1 first, ForwardIterator1 last,
                          ForwardIterator2 result);

  namespace ranges {
    template<class I, class O>
    using move_result = copy_result<I, O>;

    template<InputIterator I, Sentinel<I> S, WeaklyIncrementable O>
      requires IndirectlyMovable<I, O>
      constexpr move_result<I, O>
        move(I first, S last, O result);
    template<InputRange R, WeaklyIncrementable O>
      requires IndirectlyMovable<iterator_t<R>, O>
      constexpr move_result<safe_iterator_t<R>, O>
        move(R&& r, O result);
  }

  template<class BidirectionalIterator1, class BidirectionalIterator2>
    constexpr BidirectionalIterator2
      move_backward(BidirectionalIterator1 first, BidirectionalIterator1 last,
                    BidirectionalIterator2 result);

  namespace ranges {
    template<class I1, class I2>
    using move_backward_result = copy_result<I1, I2>;

    template<BidirectionalIterator I1, Sentinel<I1> S1, BidirectionalIterator I2>
      requires IndirectlyMovable<I1, I2>
      constexpr move_backward_result<I1, I2>
        move_backward(I1 first, S1 last, I2 result);
    template<BidirectionalRange R, BidirectionalIterator I>
      requires IndirectlyMovable<iterator_t<R>, I>
      constexpr move_backward_result<safe_iterator_t<R>, I>
        move_backward(R&& r, I result);
  }

  // \ref{alg.swap}, swap
  template<class ForwardIterator1, class ForwardIterator2>
    constexpr ForwardIterator2 swap_ranges(ForwardIterator1 first1, ForwardIterator1 last1,
                                           ForwardIterator2 first2);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2>
    ForwardIterator2 swap_ranges(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                                 ForwardIterator1 first1, ForwardIterator1 last1,
                                 ForwardIterator2 first2);

  namespace ranges {
    template<class I1, class I2>
    using swap_ranges_result = mismatch_result<I1, I2>;

    template<InputIterator I1, Sentinel<I1> S1, InputIterator I2, Sentinel<I2> S2>
      requires IndirectlySwappable<I1, I2>
      constexpr swap_ranges_result<I1, I2>
        swap_ranges(I1 first1, S1 last1, I2 first2, S2 last2);
    template<InputRange R1, InputRange R2>
      requires IndirectlySwappable<iterator_t<R1>, iterator_t<R2>>
      constexpr swap_ranges_result<safe_iterator_t<R1>, safe_iterator_t<R2>>
        swap_ranges(R1&& r1, R2&& r2);
  }

  template<class ForwardIterator1, class ForwardIterator2>
    constexpr void iter_swap(ForwardIterator1 a, ForwardIterator2 b);

  // \ref{alg.transform}, transform
  template<class InputIterator, class OutputIterator, class UnaryOperation>
    constexpr OutputIterator
      transform(InputIterator first1, InputIterator last1,
                OutputIterator result, UnaryOperation op);
  template<class InputIterator1, class InputIterator2, class OutputIterator,
           class BinaryOperation>
    constexpr OutputIterator
      transform(InputIterator1 first1, InputIterator1 last1,
                InputIterator2 first2, OutputIterator result,
                BinaryOperation binary_op);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
           class UnaryOperation>
    ForwardIterator2
      transform(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                ForwardIterator1 first1, ForwardIterator1 last1,
                ForwardIterator2 result, UnaryOperation op);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
           class ForwardIterator, class BinaryOperation>
    ForwardIterator
      transform(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                ForwardIterator1 first1, ForwardIterator1 last1,
                ForwardIterator2 first2, ForwardIterator result,
                BinaryOperation binary_op);

  namespace ranges {
    template<class I, class O>
    using unary_transform_result = copy_result<I, O>;

    template<InputIterator I, Sentinel<I> S, WeaklyIncrementable O,
             CopyConstructible F, class Proj = identity>
      requires Writable<O, indirect_result_t<F&, projected<I, Proj>>>
      constexpr unary_transform_result<I, O>
        transform(I first1, S last1, O result, F op, Proj proj = {});
    template<InputRange R, WeaklyIncrementable O, CopyConstructible F,
             class Proj = identity>
      requires Writable<O, indirect_result_t<F&, projected<iterator_t<R>, Proj>>>
      constexpr unary_transform_result<safe_iterator_t<R>, O>
        transform(R&& r, O result, F op, Proj proj = {});

    template<class I1, class I2, class O>
    struct binary_transform_result {
      [[no_unique_address]] I1 in1;
      [[no_unique_address]] I2 in2;
      [[no_unique_address]] O  out;

      template<class II1, class II2, class OO>
        requires ConvertibleTo<const I1&, II1> &&
          ConvertibleTo<const I2&, II2> && ConvertibleTo<const O&, OO>
        operator binary_transform_result<II1, II2, OO>() const & {
          return {in1, in2, out};
        }

      template<class II1, class II2, class OO>
        requires ConvertibleTo<I1, II1> &&
          ConvertibleTo<I2, II2> && ConvertibleTo<O, OO>
        operator binary_transform_result<II1, II2, OO>() && {
          return {std::move(in1), std::move(in2), std::move(out)};
        }
    };

    template<InputIterator I1, Sentinel<I1> S1, InputIterator I2, Sentinel<I2> S2,
             WeaklyIncrementable O, CopyConstructible F, class Proj1 = identity,
             class Proj2 = identity>
      requires Writable<O, indirect_result_t<F&, projected<I1, Proj1>,
                                             projected<I2, Proj2>>>
      constexpr binary_transform_result<I1, I2, O>
        transform(I1 first1, S1 last1, I2 first2, S2 last2, O result,
                  F binary_op, Proj1 proj1 = {}, Proj2 proj2 = {});
    template<InputRange R1, InputRange R2, WeaklyIncrementable O,
             CopyConstructible F, class Proj1 = identity, class Proj2 = identity>
      requires Writable<O, indirect_result_t<F&, projected<iterator_t<R1>, Proj1>,
                                             projected<iterator_t<R2>, Proj2>>>
      constexpr binary_transform_result<safe_iterator_t<R1>, safe_iterator_t<R2>, O>
        transform(R1&& r1, R2&& r2, O result,
                  F binary_op, Proj1 proj1 = {}, Proj2 proj2 = {});
  }

  // \ref{alg.replace}, replace
  template<class ForwardIterator, class T>
    constexpr void replace(ForwardIterator first, ForwardIterator last,
                           const T& old_value, const T& new_value);
  template<class ExecutionPolicy, class ForwardIterator, class T>
    void replace(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                 ForwardIterator first, ForwardIterator last,
                 const T& old_value, const T& new_value);
  template<class ForwardIterator, class Predicate, class T>
    constexpr void replace_if(ForwardIterator first, ForwardIterator last,
                              Predicate pred, const T& new_value);
  template<class ExecutionPolicy, class ForwardIterator, class Predicate, class T>
    void replace_if(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                    ForwardIterator first, ForwardIterator last,
                    Predicate pred, const T& new_value);

  namespace ranges {
    template<InputIterator I, Sentinel<I> S, class T1, class T2, class Proj = identity>
      requires Writable<I, const T2&> &&
               IndirectRelation<ranges::equal_to, projected<I, Proj>, const T1*>
      constexpr I
        replace(I first, S last, const T1& old_value, const T2& new_value, Proj proj = {});
    template<InputRange R, class T1, class T2, class Proj = identity>
      requires Writable<iterator_t<R>, const T2&> &&
               IndirectRelation<ranges::equal_to, projected<iterator_t<R>, Proj>, const T1*>
      constexpr safe_iterator_t<R>
        replace(R&& r, const T1& old_value, const T2& new_value, Proj proj = {});
    template<InputIterator I, Sentinel<I> S, class T, class Proj = identity,
             IndirectUnaryPredicate<projected<I, Proj>> Pred>
      requires Writable<I, const T&>
      constexpr I replace_if(I first, S last, Pred pred, const T& new_value, Proj proj = {});
    template<InputRange R, class T, class Proj = identity,
             IndirectUnaryPredicate<projected<iterator_t<R>, Proj>> Pred>
      requires Writable<iterator_t<R>, const T&>
      constexpr safe_iterator_t<R>
        replace_if(R&& r, Pred pred, const T& new_value, Proj proj = {});
  }

  template<class InputIterator, class OutputIterator, class T>
    constexpr OutputIterator replace_copy(InputIterator first, InputIterator last,
                                          OutputIterator result,
                                          const T& old_value, const T& new_value);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2, class T>
    ForwardIterator2 replace_copy(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                                  ForwardIterator1 first, ForwardIterator1 last,
                                  ForwardIterator2 result,
                                  const T& old_value, const T& new_value);
  template<class InputIterator, class OutputIterator, class Predicate, class T>
    constexpr OutputIterator replace_copy_if(InputIterator first, InputIterator last,
                                             OutputIterator result,
                                             Predicate pred, const T& new_value);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
           class Predicate, class T>
    ForwardIterator2 replace_copy_if(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                                     ForwardIterator1 first, ForwardIterator1 last,
                                     ForwardIterator2 result,
                                     Predicate pred, const T& new_value);

  namespace ranges {
    template<class I, class O>
    using replace_copy_result = copy_result<I, O>;

    template<InputIterator I, Sentinel<I> S, class T1, class T2, OutputIterator<const T2&> O,
             class Proj = identity>
      requires IndirectlyCopyable<I, O> &&
               IndirectRelation<ranges::equal_to, projected<I, Proj>, const T1*>
      constexpr replace_copy_result<I, O>
        replace_copy(I first, S last, O result, const T1& old_value, const T2& new_value,
                     Proj proj = {});
    template<InputRange R, class T1, class T2, OutputIterator<const T2&> O,
             class Proj = identity>
      requires IndirectlyCopyable<iterator_t<R>, O> &&
               IndirectRelation<ranges::equal_to, projected<iterator_t<R>, Proj>, const T1*>
      constexpr replace_copy_result<safe_iterator_t<R>, O>
        replace_copy(R&& r, O result, const T1& old_value, const T2& new_value,
                     Proj proj = {});

    template<class I, class O>
    using replace_copy_if_result = copy_result<I, O>;

    template<InputIterator I, Sentinel<I> S, class T, OutputIterator<const T&> O,
             class Proj = identity, IndirectUnaryPredicate<projected<I, Proj>> Pred>
      requires IndirectlyCopyable<I, O>
      constexpr replace_copy_if_result<I, O>
        replace_copy_if(I first, S last, O result, Pred pred, const T& new_value,
                        Proj proj = {});
    template<InputRange R, class T, OutputIterator<const T&> O, class Proj = identity,
             IndirectUnaryPredicate<projected<iterator_t<R>, Proj>> Pred>
      requires IndirectlyCopyable<iterator_t<R>, O>
      constexpr replace_copy_if_result<safe_iterator_t<R>, O>
        replace_copy_if(R&& r, O result, Pred pred, const T& new_value,
                        Proj proj = {});
  }

  // \ref{alg.fill}, fill
  template<class ForwardIterator, class T>
    constexpr void fill(ForwardIterator first, ForwardIterator last, const T& value);
  template<class ExecutionPolicy, class ForwardIterator, class T>
    void fill(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
              ForwardIterator first, ForwardIterator last, const T& value);
  template<class OutputIterator, class Size, class T>
    constexpr OutputIterator fill_n(OutputIterator first, Size n, const T& value);
  template<class ExecutionPolicy, class ForwardIterator,
           class Size, class T>
    ForwardIterator fill_n(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                           ForwardIterator first, Size n, const T& value);

  namespace ranges {
    template<class T, OutputIterator<const T&> O, Sentinel<O> S>
      constexpr O fill(O first, S last, const T& value);
    template<class T, OutputRange<const T&> R>
      constexpr safe_iterator_t<R> fill(R&& r, const T& value);
    template<class T, OutputIterator<const T&> O>
      constexpr O fill_n(O first, iter_difference_t<O> n, const T& value);
  }

  // \ref{alg.generate}, generate
  template<class ForwardIterator, class Generator>
    constexpr void generate(ForwardIterator first, ForwardIterator last,
                            Generator gen);
  template<class ExecutionPolicy, class ForwardIterator, class Generator>
    void generate(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                  ForwardIterator first, ForwardIterator last,
                  Generator gen);
  template<class OutputIterator, class Size, class Generator>
    constexpr OutputIterator generate_n(OutputIterator first, Size n, Generator gen);
  template<class ExecutionPolicy, class ForwardIterator, class Size, class Generator>
    ForwardIterator generate_n(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                               ForwardIterator first, Size n, Generator gen);

  namespace ranges {
    template<Iterator O, Sentinel<O> S, CopyConstructible F>
      requires Invocable<F&> && Writable<O, invoke_result_t<F&>>
      constexpr O generate(O first, S last, F gen);
    template<class R, CopyConstructible F>
      requires Invocable<F&> && OutputRange<R, invoke_result_t<F&>>
      constexpr safe_iterator_t<R> generate(R&& r, F gen);
    template<Iterator O, CopyConstructible F>
      requires Invocable<F&> && Writable<O, invoke_result_t<F&>>
      constexpr O generate_n(O first, iter_difference_t<O> n, F gen);
  }

  // \ref{alg.remove}, remove
  template<class ForwardIterator, class T>
    constexpr ForwardIterator remove(ForwardIterator first, ForwardIterator last,
                                     const T& value);
  template<class ExecutionPolicy, class ForwardIterator, class T>
    ForwardIterator remove(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                           ForwardIterator first, ForwardIterator last,
                           const T& value);
  template<class ForwardIterator, class Predicate>
    constexpr ForwardIterator remove_if(ForwardIterator first, ForwardIterator last,
                                        Predicate pred);
  template<class ExecutionPolicy, class ForwardIterator, class Predicate>
    ForwardIterator remove_if(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                              ForwardIterator first, ForwardIterator last,
                              Predicate pred);

  namespace ranges {
    template<Permutable I, Sentinel<I> S, class T, class Proj = identity>
      requires IndirectRelation<ranges::equal_to, projected<I, Proj>, const T*>
      constexpr I remove(I first, S last, const T& value, Proj proj = {});
    template<ForwardRange R, class T, class Proj = identity>
      requires Permutable<iterator_t<R>> &&
               IndirectRelation<ranges::equal_to, projected<iterator_t<R>, Proj>, const T*>
      constexpr safe_iterator_t<R>
        remove(R&& r, const T& value, Proj proj = {});
    template<Permutable I, Sentinel<I> S, class Proj = identity,
             IndirectUnaryPredicate<projected<I, Proj>> Pred>
      constexpr I remove_if(I first, S last, Pred pred, Proj proj = {});
    template<ForwardRange R, class Proj = identity,
             IndirectUnaryPredicate<projected<iterator_t<R>, Proj>> Pred>
      requires Permutable<iterator_t<R>>
      constexpr safe_iterator_t<R>
        remove_if(R&& r, Pred pred, Proj proj = {});
  }

  template<class InputIterator, class OutputIterator, class T>
    constexpr OutputIterator
      remove_copy(InputIterator first, InputIterator last,
                  OutputIterator result, const T& value);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
           class T>
    ForwardIterator2
      remove_copy(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                  ForwardIterator1 first, ForwardIterator1 last,
                  ForwardIterator2 result, const T& value);
  template<class InputIterator, class OutputIterator, class Predicate>
    constexpr OutputIterator
      remove_copy_if(InputIterator first, InputIterator last,
                     OutputIterator result, Predicate pred);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
           class Predicate>
    ForwardIterator2
      remove_copy_if(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                     ForwardIterator1 first, ForwardIterator1 last,
                     ForwardIterator2 result, Predicate pred);

  namespace ranges {
    template<class I, class O>
    using remove_copy_result = copy_result<I, O>;

    template<InputIterator I, Sentinel<I> S, WeaklyIncrementable O, class T,
             class Proj = identity>
      requires IndirectlyCopyable<I, O> &&
               IndirectRelation<ranges::equal_to, projected<I, Proj>, const T*>
      constexpr remove_copy_result<I, O>
        remove_copy(I first, S last, O result, const T& value, Proj proj = {});
    template<InputRange R, WeaklyIncrementable O, class T, class Proj = identity>
      requires IndirectlyCopyable<iterator_t<R>, O> &&
               IndirectRelation<ranges::equal_to, projected<iterator_t<R>, Proj>, const T*>
      constexpr remove_copy_result<safe_iterator_t<R>, O>
        remove_copy(R&& r, O result, const T& value, Proj proj = {});

    template<class I, class O>
    using remove_copy_if_result = copy_result<I, O>;

    template<InputIterator I, Sentinel<I> S, WeaklyIncrementable O,
             class Proj = identity, IndirectUnaryPredicate<projected<I, Proj>> Pred>
      requires IndirectlyCopyable<I, O>
      constexpr remove_copy_if_result<I, O>
        remove_copy_if(I first, S last, O result, Pred pred, Proj proj = {});
    template<InputRange R, WeaklyIncrementable O, class Proj = identity,
             IndirectUnaryPredicate<projected<iterator_t<R>, Proj>> Pred>
      requires IndirectlyCopyable<iterator_t<R>, O>
      constexpr remove_copy_if_result<safe_iterator_t<R>, O>
        remove_copy_if(R&& r, O result, Pred pred, Proj proj = {});
  }

  // \ref{alg.unique}, unique
  template<class ForwardIterator>
    constexpr ForwardIterator unique(ForwardIterator first, ForwardIterator last);
  template<class ForwardIterator, class BinaryPredicate>
    constexpr ForwardIterator unique(ForwardIterator first, ForwardIterator last,
                                     BinaryPredicate pred);
  template<class ExecutionPolicy, class ForwardIterator>
    ForwardIterator unique(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                           ForwardIterator first, ForwardIterator last);
  template<class ExecutionPolicy, class ForwardIterator, class BinaryPredicate>
    ForwardIterator unique(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                           ForwardIterator first, ForwardIterator last,
                           BinaryPredicate pred);

  namespace ranges {
    template<Permutable I, Sentinel<I> S, class Proj = identity,
             IndirectRelation<projected<I, Proj>> C = ranges::equal_to>
      constexpr I unique(I first, S last, C comp = {}, Proj proj = {});
    template<ForwardRange R, class Proj = identity,
             IndirectRelation<projected<iterator_t<R>, Proj>> C = ranges::equal_to>
      requires Permutable<iterator_t<R>>
      constexpr safe_iterator_t<R>
        unique(R&& r, C comp = {}, Proj proj = {});
  }

  template<class InputIterator, class OutputIterator>
    constexpr OutputIterator
      unique_copy(InputIterator first, InputIterator last,
                  OutputIterator result);
  template<class InputIterator, class OutputIterator, class BinaryPredicate>
    constexpr OutputIterator
      unique_copy(InputIterator first, InputIterator last,
                  OutputIterator result, BinaryPredicate pred);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2>
    ForwardIterator2
      unique_copy(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                  ForwardIterator1 first, ForwardIterator1 last,
                  ForwardIterator2 result);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
           class BinaryPredicate>
    ForwardIterator2
      unique_copy(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                  ForwardIterator1 first, ForwardIterator1 last,
                  ForwardIterator2 result, BinaryPredicate pred);

  namespace ranges {
    template<class I, class O>
    using unique_copy_result = copy_result<I, O>;

    template<InputIterator I, Sentinel<I> S, WeaklyIncrementable O,
             class Proj = identity, IndirectRelation<projected<I, Proj>> C = ranges::equal_to>
      requires IndirectlyCopyable<I, O> &&
               (ForwardIterator<I> ||
                (InputIterator<O> && Same<iter_value_t<I>, iter_value_t<O>>) ||
                IndirectlyCopyableStorable<I, O>)
      constexpr unique_copy_result<I, O>
        unique_copy(I first, S last, O result, C comp = {}, Proj proj = {});
    template<InputRange R, WeaklyIncrementable O, class Proj = identity,
             IndirectRelation<projected<iterator_t<R>, Proj>> C = ranges::equal_to>
      requires IndirectlyCopyable<iterator_t<R>, O> &&
               (ForwardIterator<iterator_t<R>> ||
                (InputIterator<O> && Same<iter_value_t<iterator_t<R>>, iter_value_t<O>>) ||
                IndirectlyCopyableStorable<iterator_t<R>, O>)
      constexpr unique_copy_result<safe_iterator_t<R>, O>
        unique_copy(R&& r, O result, C comp = {}, Proj proj = {});
  }

  // \ref{alg.reverse}, reverse
  template<class BidirectionalIterator>
    constexpr void reverse(BidirectionalIterator first, BidirectionalIterator last);
  template<class ExecutionPolicy, class BidirectionalIterator>
    void reverse(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                 BidirectionalIterator first, BidirectionalIterator last);

  namespace ranges {
    template<BidirectionalIterator I, Sentinel<I> S>
      requires Permutable<I>
      constexpr I reverse(I first, S last);
    template<BidirectionalRange R>
      requires Permutable<iterator_t<R>>
      constexpr safe_iterator_t<R> reverse(R&& r);
  }

  template<class BidirectionalIterator, class OutputIterator>
    constexpr OutputIterator
      reverse_copy(BidirectionalIterator first, BidirectionalIterator last,
                   OutputIterator result);
  template<class ExecutionPolicy, class BidirectionalIterator, class ForwardIterator>
    ForwardIterator
      reverse_copy(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                   BidirectionalIterator first, BidirectionalIterator last,
                   ForwardIterator result);

  namespace ranges {
    template<class I, class O>
    using reverse_copy_result = copy_result<I, O>;

    template<BidirectionalIterator I, Sentinel<I> S, WeaklyIncrementable O>
      requires IndirectlyCopyable<I, O>
      constexpr reverse_copy_result<I, O>
        reverse_copy(I first, S last, O result);
    template<BidirectionalRange R, WeaklyIncrementable O>
      requires IndirectlyCopyable<iterator_t<R>, O>
      constexpr reverse_copy_result<safe_iterator_t<R>, O>
        reverse_copy(R&& r, O result);
  }

  // \ref{alg.rotate}, rotate
  template<class ForwardIterator>
    constexpr ForwardIterator rotate(ForwardIterator first,
                                     ForwardIterator middle,
                                     ForwardIterator last);
  template<class ExecutionPolicy, class ForwardIterator>
    ForwardIterator rotate(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                           ForwardIterator first,
                           ForwardIterator middle,
                           ForwardIterator last);

  namespace ranges {
    template<Permutable I, Sentinel<I> S>
      constexpr subrange<I> rotate(I first, I middle, S last);
    template<ForwardRange R>
      requires Permutable<iterator_t<R>>
      constexpr safe_subrange_t<R> rotate(R&& r, iterator_t<R> middle);
  }

  template<class ForwardIterator, class OutputIterator>
    constexpr OutputIterator
      rotate_copy(ForwardIterator first, ForwardIterator middle,
                  ForwardIterator last, OutputIterator result);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2>
    ForwardIterator2
      rotate_copy(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                  ForwardIterator1 first, ForwardIterator1 middle,
                  ForwardIterator1 last, ForwardIterator2 result);

  namespace ranges {
    template<class I, class O>
    using rotate_copy_result = copy_result<I, O>;

    template<ForwardIterator I, Sentinel<I> S, WeaklyIncrementable O>
      requires IndirectlyCopyable<I, O>
      constexpr rotate_copy_result<I, O>
        rotate_copy(I first, I middle, S last, O result);
    template<ForwardRange R, WeaklyIncrementable O>
      requires IndirectlyCopyable<iterator_t<R>, O>
      constexpr rotate_copy_result<safe_iterator_t<R>, O>
        rotate_copy(R&& r, iterator_t<R> middle, O result);
  }

  // \ref{alg.random.sample}, sample
  template<class PopulationIterator, class SampleIterator,
           class Distance, class UniformRandomBitGenerator>
    SampleIterator sample(PopulationIterator first, PopulationIterator last,
                          SampleIterator out, Distance n,
                          UniformRandomBitGenerator&& g);

  // \ref{alg.random.shuffle}, shuffle
  template<class RandomAccessIterator, class UniformRandomBitGenerator>
    void shuffle(RandomAccessIterator first,
                 RandomAccessIterator last,
                 UniformRandomBitGenerator&& g);

  namespace ranges {
    template<RandomAccessIterator I, Sentinel<I> S, class Gen>
      requires Permutable<I> &&
               UniformRandomBitGenerator<remove_reference_t<Gen>> &&
               ConvertibleTo<invoke_result_t<Gen&>, iter_difference_t<I>>
      I shuffle(I first, S last, Gen&& g);
    template<RandomAccessRange R, class Gen>
      requires Permutable<iterator_t<R>> &&
               UniformRandomBitGenerator<remove_reference_t<Gen>> &&
               ConvertibleTo<invoke_result_t<Gen&>, iter_difference_t<iterator_t<R>>>
      safe_iterator_t<R> shuffle(R&& r, Gen&& g);
  }

  // \ref{alg.shift}, shift
  template<class ForwardIterator>
    constexpr ForwardIterator
      shift_left(ForwardIterator first, ForwardIterator last,
                 typename iterator_traits<ForwardIterator>::difference_type n);
  template<class ExecutionPolicy, class ForwardIterator>
    ForwardIterator
      shift_left(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                 ForwardIterator first, ForwardIterator last,
                 typename iterator_traits<ForwardIterator>::difference_type n);
  template<class ForwardIterator>
    constexpr ForwardIterator
      shift_right(ForwardIterator first, ForwardIterator last,
                  typename iterator_traits<ForwardIterator>::difference_type n);
  template<class ExecutionPolicy, class ForwardIterator>
    ForwardIterator
      shift_right(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                  ForwardIterator first, ForwardIterator last,
                  typename iterator_traits<ForwardIterator>::difference_type n);

  // \ref{alg.sorting}, sorting and related operations
  // \ref{alg.sort}, sorting
  template<class RandomAccessIterator>
    constexpr void sort(RandomAccessIterator first, RandomAccessIterator last);
  template<class RandomAccessIterator, class Compare>
    constexpr void sort(RandomAccessIterator first, RandomAccessIterator last,
                        Compare comp);
  template<class ExecutionPolicy, class RandomAccessIterator>
    void sort(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
              RandomAccessIterator first, RandomAccessIterator last);
  template<class ExecutionPolicy, class RandomAccessIterator, class Compare>
    void sort(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
              RandomAccessIterator first, RandomAccessIterator last,
              Compare comp);

  namespace ranges {
    template<RandomAccessIterator I, Sentinel<I> S, class Comp = ranges::less,
             class Proj = identity>
      requires Sortable<I, Comp, Proj>
      constexpr I
        sort(I first, S last, Comp comp = {}, Proj proj = {});
    template<RandomAccessRange R, class Comp = ranges::less, class Proj = identity>
      requires Sortable<iterator_t<R>, Comp, Proj>
      constexpr safe_iterator_t<R>
        sort(R&& r, Comp comp = {}, Proj proj = {});
  }

  template<class RandomAccessIterator>
    void stable_sort(RandomAccessIterator first, RandomAccessIterator last);
  template<class RandomAccessIterator, class Compare>
    void stable_sort(RandomAccessIterator first, RandomAccessIterator last,
                     Compare comp);
  template<class ExecutionPolicy, class RandomAccessIterator>
    void stable_sort(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                     RandomAccessIterator first, RandomAccessIterator last);
  template<class ExecutionPolicy, class RandomAccessIterator, class Compare>
    void stable_sort(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                     RandomAccessIterator first, RandomAccessIterator last,
                     Compare comp);

  namespace ranges {
    template<RandomAccessIterator I, Sentinel<I> S, class Comp = ranges::less,
             class Proj = identity>
      requires Sortable<I, Comp, Proj>
      I stable_sort(I first, S last, Comp comp = {}, Proj proj = {});
    template<RandomAccessRange R, class Comp = ranges::less, class Proj = identity>
      requires Sortable<iterator_t<R>, Comp, Proj>
      safe_iterator_t<R>
        stable_sort(R&& r, Comp comp = {}, Proj proj = {});
  }

  template<class RandomAccessIterator>
    constexpr void partial_sort(RandomAccessIterator first,
                                RandomAccessIterator middle,
                                RandomAccessIterator last);
  template<class RandomAccessIterator, class Compare>
    constexpr void partial_sort(RandomAccessIterator first,
                                RandomAccessIterator middle,
                                RandomAccessIterator last, Compare comp);
  template<class ExecutionPolicy, class RandomAccessIterator>
    void partial_sort(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                      RandomAccessIterator first,
                      RandomAccessIterator middle,
                      RandomAccessIterator last);
  template<class ExecutionPolicy, class RandomAccessIterator, class Compare>
    void partial_sort(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                      RandomAccessIterator first,
                      RandomAccessIterator middle,
                      RandomAccessIterator last, Compare comp);

  namespace ranges {
    template<RandomAccessIterator I, Sentinel<I> S, class Comp = ranges::less,
             class Proj = identity>
      requires Sortable<I, Comp, Proj>
      constexpr I
        partial_sort(I first, I middle, S last, Comp comp = {}, Proj proj = {});
    template<RandomAccessRange R, class Comp = ranges::less, class Proj = identity>
      requires Sortable<iterator_t<R>, Comp, Proj>
      constexpr safe_iterator_t<R>
        partial_sort(R&& r, iterator_t<R> middle, Comp comp = {},
                     Proj proj = {});
  }

  template<class InputIterator, class RandomAccessIterator>
    constexpr RandomAccessIterator
      partial_sort_copy(InputIterator first, InputIterator last,
                        RandomAccessIterator result_first,
                        RandomAccessIterator result_last);
  template<class InputIterator, class RandomAccessIterator, class Compare>
    constexpr RandomAccessIterator
      partial_sort_copy(InputIterator first, InputIterator last,
                        RandomAccessIterator result_first,
                        RandomAccessIterator result_last,
                        Compare comp);
  template<class ExecutionPolicy, class ForwardIterator, class RandomAccessIterator>
    RandomAccessIterator
      partial_sort_copy(ExecutionPolicy&& exec,  // see \ref{algorithms.parallel.overloads}
                        ForwardIterator first, ForwardIterator last,
                        RandomAccessIterator result_first,
                        RandomAccessIterator result_last);
  template<class ExecutionPolicy, class ForwardIterator, class RandomAccessIterator,
           class Compare>
    RandomAccessIterator
      partial_sort_copy(ExecutionPolicy&& exec,  // see \ref{algorithms.parallel.overloads}
                        ForwardIterator first, ForwardIterator last,
                        RandomAccessIterator result_first,
                        RandomAccessIterator result_last,
                        Compare comp);

  namespace ranges {
    template<InputIterator I1, Sentinel<I1> S1, RandomAccessIterator I2, Sentinel<I2> S2,
             class Comp = ranges::less, class Proj1 = identity, class Proj2 = identity>
      requires IndirectlyCopyable<I1, I2> && Sortable<I2, Comp, Proj2> &&
               IndirectStrictWeakOrder<Comp, projected<I1, Proj1>, projected<I2, Proj2>>
      constexpr I2
        partial_sort_copy(I1 first, S1 last, I2 result_first, S2 result_last,
                          Comp comp = {}, Proj1 proj1 = {}, Proj2 proj2 = {});
    template<InputRange R1, RandomAccessRange R2, class Comp = ranges::less,
             class Proj1 = identity, class Proj2 = identity>
      requires IndirectlyCopyable<iterator_t<R1>, iterator_t<R2>> &&
               Sortable<iterator_t<R2>, Comp, Proj2> &&
               IndirectStrictWeakOrder<Comp, projected<iterator_t<R1>, Proj1>,
                                       projected<iterator_t<R2>, Proj2>>
      constexpr safe_iterator_t<R2>
        partial_sort_copy(R1&& r, R2&& result_r, Comp comp = {},
                          Proj1 proj1 = {}, Proj2 proj2 = {});
  }

  template<class ForwardIterator>
    constexpr bool is_sorted(ForwardIterator first, ForwardIterator last);
  template<class ForwardIterator, class Compare>
    constexpr bool is_sorted(ForwardIterator first, ForwardIterator last,
                             Compare comp);
  template<class ExecutionPolicy, class ForwardIterator>
    bool is_sorted(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                   ForwardIterator first, ForwardIterator last);
  template<class ExecutionPolicy, class ForwardIterator, class Compare>
    bool is_sorted(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                   ForwardIterator first, ForwardIterator last,
                   Compare comp);

  namespace ranges {
    template<ForwardIterator I, Sentinel<I> S, class Proj = identity,
             IndirectStrictWeakOrder<projected<I, Proj>> Comp = ranges::less>
      constexpr bool is_sorted(I first, S last, Comp comp = {}, Proj proj = {});
    template<ForwardRange R, class Proj = identity,
             IndirectStrictWeakOrder<projected<iterator_t<R>, Proj>> Comp = ranges::less>
      constexpr bool is_sorted(R&& r, Comp comp = {}, Proj proj = {});
  }

  template<class ForwardIterator>
    constexpr ForwardIterator
      is_sorted_until(ForwardIterator first, ForwardIterator last);
  template<class ForwardIterator, class Compare>
    constexpr ForwardIterator
      is_sorted_until(ForwardIterator first, ForwardIterator last,
                      Compare comp);
  template<class ExecutionPolicy, class ForwardIterator>
    ForwardIterator
      is_sorted_until(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                      ForwardIterator first, ForwardIterator last);
  template<class ExecutionPolicy, class ForwardIterator, class Compare>
    ForwardIterator
      is_sorted_until(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                      ForwardIterator first, ForwardIterator last,
                      Compare comp);

  namespace ranges {
    template<ForwardIterator I, Sentinel<I> S, class Proj = identity,
             IndirectStrictWeakOrder<projected<I, Proj>> Comp = ranges::less>
      constexpr I is_sorted_until(I first, S last, Comp comp = {}, Proj proj = {});
    template<ForwardRange R, class Proj = identity,
             IndirectStrictWeakOrder<projected<iterator_t<R>, Proj>> Comp = ranges::less>
      constexpr safe_iterator_t<R>
        is_sorted_until(R&& r, Comp comp = {}, Proj proj = {});
  }

  // \ref{alg.nth.element}, Nth element
  template<class RandomAccessIterator>
    constexpr void nth_element(RandomAccessIterator first, RandomAccessIterator nth,
                               RandomAccessIterator last);
  template<class RandomAccessIterator, class Compare>
    constexpr void nth_element(RandomAccessIterator first, RandomAccessIterator nth,
                               RandomAccessIterator last, Compare comp);
  template<class ExecutionPolicy, class RandomAccessIterator>
    void nth_element(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                     RandomAccessIterator first, RandomAccessIterator nth,
                     RandomAccessIterator last);
  template<class ExecutionPolicy, class RandomAccessIterator, class Compare>
    void nth_element(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                     RandomAccessIterator first, RandomAccessIterator nth,
                     RandomAccessIterator last, Compare comp);

  namespace ranges {
    template<RandomAccessIterator I, Sentinel<I> S, class Comp = ranges::less,
             class Proj = identity>
      requires Sortable<I, Comp, Proj>
      constexpr I
        nth_element(I first, I nth, S last, Comp comp = {}, Proj proj = {});
    template<RandomAccessRange R, class Comp = ranges::less, class Proj = identity>
      requires Sortable<iterator_t<R>, Comp, Proj>
      constexpr safe_iterator_t<R>
        nth_element(R&& r, iterator_t<R> nth, Comp comp = {}, Proj proj = {});
  }

  // \ref{alg.binary.search}, binary search
  template<class ForwardIterator, class T>
    constexpr ForwardIterator
      lower_bound(ForwardIterator first, ForwardIterator last,
                  const T& value);
  template<class ForwardIterator, class T, class Compare>
    constexpr ForwardIterator
      lower_bound(ForwardIterator first, ForwardIterator last,
                  const T& value, Compare comp);

  namespace ranges {
    template<ForwardIterator I, Sentinel<I> S, class T, class Proj = identity,
             IndirectStrictWeakOrder<const T*, projected<I, Proj>> Comp = ranges::less>
      constexpr I lower_bound(I first, S last, const T& value, Comp comp = {},
                              Proj proj = {});
    template<ForwardRange R, class T, class Proj = identity,
             IndirectStrictWeakOrder<const T*, projected<iterator_t<R>, Proj>> Comp =
               ranges::less>
      constexpr safe_iterator_t<R>
        lower_bound(R&& r, const T& value, Comp comp = {}, Proj proj = {});
  }

  template<class ForwardIterator, class T>
    constexpr ForwardIterator
      upper_bound(ForwardIterator first, ForwardIterator last,
                  const T& value);
  template<class ForwardIterator, class T, class Compare>
    constexpr ForwardIterator
      upper_bound(ForwardIterator first, ForwardIterator last,
                  const T& value, Compare comp);

  namespace ranges {
    template<ForwardIterator I, Sentinel<I> S, class T, class Proj = identity,
             IndirectStrictWeakOrder<const T*, projected<I, Proj>> Comp = ranges::less>
      constexpr I upper_bound(I first, S last, const T& value, Comp comp = {}, Proj proj = {});
    template<ForwardRange R, class T, class Proj = identity,
             IndirectStrictWeakOrder<const T*, projected<iterator_t<R>, Proj>> Comp =
               ranges::less>
      constexpr safe_iterator_t<R>
        upper_bound(R&& r, const T& value, Comp comp = {}, Proj proj = {});
  }

  template<class ForwardIterator, class T>
    constexpr pair<ForwardIterator, ForwardIterator>
      equal_range(ForwardIterator first, ForwardIterator last,
                  const T& value);
  template<class ForwardIterator, class T, class Compare>
    constexpr pair<ForwardIterator, ForwardIterator>
      equal_range(ForwardIterator first, ForwardIterator last,
                  const T& value, Compare comp);

  namespace ranges {
    template<ForwardIterator I, Sentinel<I> S, class T, class Proj = identity,
             IndirectStrictWeakOrder<const T*, projected<I, Proj>> Comp = ranges::less>
      constexpr subrange<I>
        equal_range(I first, S last, const T& value, Comp comp = {}, Proj proj = {});
    template<ForwardRange R, class T, class Proj = identity,
             IndirectStrictWeakOrder<const T*, projected<iterator_t<R>, Proj>> Comp =
               ranges::less>
      constexpr safe_subrange_t<R>
        equal_range(R&& r, const T& value, Comp comp = {}, Proj proj = {});
  }

  template<class ForwardIterator, class T>
    constexpr bool
      binary_search(ForwardIterator first, ForwardIterator last,
                    const T& value);
  template<class ForwardIterator, class T, class Compare>
    constexpr bool
      binary_search(ForwardIterator first, ForwardIterator last,
                    const T& value, Compare comp);

  namespace ranges {
    template<ForwardIterator I, Sentinel<I> S, class T, class Proj = identity,
             IndirectStrictWeakOrder<const T*, projected<I, Proj>> Comp = ranges::less>
      constexpr bool binary_search(I first, S last, const T& value, Comp comp = {},
                                   Proj proj = {});
    template<ForwardRange R, class T, class Proj = identity,
             IndirectStrictWeakOrder<const T*, projected<iterator_t<R>, Proj>> Comp =
               ranges::less>
      constexpr bool binary_search(R&& r, const T& value, Comp comp = {},
                                   Proj proj = {});
  }

  // \ref{alg.partitions}, partitions
  template<class InputIterator, class Predicate>
    constexpr bool is_partitioned(InputIterator first, InputIterator last, Predicate pred);
  template<class ExecutionPolicy, class ForwardIterator, class Predicate>
    bool is_partitioned(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                        ForwardIterator first, ForwardIterator last, Predicate pred);

  namespace ranges {
    template<InputIterator I, Sentinel<I> S, class Proj = identity,
             IndirectUnaryPredicate<projected<I, Proj>> Pred>
      constexpr bool is_partitioned(I first, S last, Pred pred, Proj proj = {});
    template<InputRange R, class Proj = identity,
             IndirectUnaryPredicate<projected<iterator_t<R>, Proj>> Pred>
      constexpr bool is_partitioned(R&& r, Pred pred, Proj proj = {});
  }

  template<class ForwardIterator, class Predicate>
    constexpr ForwardIterator partition(ForwardIterator first,
                                        ForwardIterator last,
                                        Predicate pred);
  template<class ExecutionPolicy, class ForwardIterator, class Predicate>
    ForwardIterator partition(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                              ForwardIterator first,
                              ForwardIterator last,
                              Predicate pred);

  namespace ranges {
    template<Permutable I, Sentinel<I> S, class Proj = identity,
             IndirectUnaryPredicate<projected<I, Proj>> Pred>
      constexpr I
        partition(I first, S last, Pred pred, Proj proj = {});
    template<ForwardRange R, class Proj = identity,
             IndirectUnaryPredicate<projected<iterator_t<R>, Proj>> Pred>
      requires Permutable<iterator_t<R>>
      constexpr safe_iterator_t<R>
        partition(R&& r, Pred pred, Proj proj = {});
  }

  template<class BidirectionalIterator, class Predicate>
    BidirectionalIterator stable_partition(BidirectionalIterator first,
                                           BidirectionalIterator last,
                                           Predicate pred);
  template<class ExecutionPolicy, class BidirectionalIterator, class Predicate>
    BidirectionalIterator stable_partition(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                                           BidirectionalIterator first,
                                           BidirectionalIterator last,
                                           Predicate pred);

  namespace ranges {
    template<BidirectionalIterator I, Sentinel<I> S, class Proj = identity,
             IndirectUnaryPredicate<projected<I, Proj>> Pred>
      requires Permutable<I>
      I stable_partition(I first, S last, Pred pred, Proj proj = {});
    template<BidirectionalRange R, class Proj = identity,
             IndirectUnaryPredicate<projected<iterator_t<R>, Proj>> Pred>
      requires Permutable<iterator_t<R>>
      safe_iterator_t<R> stable_partition(R&& r, Pred pred, Proj proj = {});
  }

  template<class InputIterator, class OutputIterator1,
           class OutputIterator2, class Predicate>
    constexpr pair<OutputIterator1, OutputIterator2>
      partition_copy(InputIterator first, InputIterator last,
                     OutputIterator1 out_true, OutputIterator2 out_false,
                     Predicate pred);
  template<class ExecutionPolicy, class ForwardIterator, class ForwardIterator1,
           class ForwardIterator2, class Predicate>
    pair<ForwardIterator1, ForwardIterator2>
      partition_copy(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                     ForwardIterator first, ForwardIterator last,
                     ForwardIterator1 out_true, ForwardIterator2 out_false,
                     Predicate pred);

  namespace ranges {
    template<class I, class O1, class O2>
    struct partition_copy_result {
      [[no_unique_address]] I  in;
      [[no_unique_address]] O1 out1;
      [[no_unique_address]] O2 out2;

      template<class II, class OO1, class OO2>
        requires ConvertibleTo<const I&, II> &&
          ConvertibleTo<const O1&, OO1> && ConvertibleTo<const O2&, OO2>
        operator partition_copy_result<II, OO1, OO2>() const & {
          return {in, out1, out2};
        }

      template<class II, class OO1, class OO2>
        requires ConvertibleTo<I, II> &&
          ConvertibleTo<O1, OO1> && ConvertibleTo<O2, OO2>
        operator partition_copy_result<II, OO1, OO2>() && {
          return {std::move(in), std::move(out1), std::move(out2)};
        }
    };

    template<InputIterator I, Sentinel<I> S, WeaklyIncrementable O1, WeaklyIncrementable O2,
             class Proj = identity, IndirectUnaryPredicate<projected<I, Proj>> Pred>
      requires IndirectlyCopyable<I, O1> && IndirectlyCopyable<I, O2>
      constexpr partition_copy_result<I, O1, O2>
        partition_copy(I first, S last, O1 out_true, O2 out_false, Pred pred,
                       Proj proj = {});
    template<InputRange R, WeaklyIncrementable O1, WeaklyIncrementable O2,
             class Proj = identity,
             IndirectUnaryPredicate<projected<iterator_t<R>, Proj>> Pred>
      requires IndirectlyCopyable<iterator_t<R>, O1> &&
               IndirectlyCopyable<iterator_t<R>, O2>
      constexpr partition_copy_result<safe_iterator_t<R>, O1, O2>
        partition_copy(R&& r, O1 out_true, O2 out_false, Pred pred, Proj proj = {});
  }

  template<class ForwardIterator, class Predicate>
    constexpr ForwardIterator
      partition_point(ForwardIterator first, ForwardIterator last,
                      Predicate pred);

  namespace ranges {
    template<ForwardIterator I, Sentinel<I> S, class Proj = identity,
             IndirectUnaryPredicate<projected<I, Proj>> Pred>
      constexpr I partition_point(I first, S last, Pred pred, Proj proj = {});
    template<ForwardRange R, class Proj = identity,
             IndirectUnaryPredicate<projected<iterator_t<R>, Proj>> Pred>
      constexpr safe_iterator_t<R>
        partition_point(R&& r, Pred pred, Proj proj = {});
  }

  // \ref{alg.merge}, merge
  template<class InputIterator1, class InputIterator2, class OutputIterator>
    constexpr OutputIterator
      merge(InputIterator1 first1, InputIterator1 last1,
            InputIterator2 first2, InputIterator2 last2,
            OutputIterator result);
  template<class InputIterator1, class InputIterator2, class OutputIterator,
           class Compare>
    constexpr OutputIterator
      merge(InputIterator1 first1, InputIterator1 last1,
            InputIterator2 first2, InputIterator2 last2,
            OutputIterator result, Compare comp);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
           class ForwardIterator>
    ForwardIterator
      merge(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
            ForwardIterator1 first1, ForwardIterator1 last1,
            ForwardIterator2 first2, ForwardIterator2 last2,
            ForwardIterator result);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
           class ForwardIterator, class Compare>
    ForwardIterator
      merge(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
            ForwardIterator1 first1, ForwardIterator1 last1,
            ForwardIterator2 first2, ForwardIterator2 last2,
            ForwardIterator result, Compare comp);

  namespace ranges {
    template<class I1, class I2, class O>
    using merge_result = binary_transform_result<I1, I2, O>;

    template<InputIterator I1, Sentinel<I1> S1, InputIterator I2, Sentinel<I2> S2,
             WeaklyIncrementable O, class Comp = ranges::less, class Proj1 = identity,
             class Proj2 = identity>
      requires Mergeable<I1, I2, O, Comp, Proj1, Proj2>
      constexpr merge_result<I1, I2, O>
        merge(I1 first1, S1 last1, I2 first2, S2 last2, O result,
              Comp comp = {}, Proj1 proj1 = {}, Proj2 proj2 = {});
    template<InputRange R1, InputRange R2, WeaklyIncrementable O, class Comp = ranges::less,
             class Proj1 = identity, class Proj2 = identity>
      requires Mergeable<iterator_t<R1>, iterator_t<R2>, O, Comp, Proj1, Proj2>
      constexpr merge_result<safe_iterator_t<R1>, safe_iterator_t<R2>, O>
        merge(R1&& r1, R2&& r2, O result,
              Comp comp = {}, Proj1 proj1 = {}, Proj2 proj2 = {});
  }

  template<class BidirectionalIterator>
    void inplace_merge(BidirectionalIterator first,
                       BidirectionalIterator middle,
                       BidirectionalIterator last);
  template<class BidirectionalIterator, class Compare>
    void inplace_merge(BidirectionalIterator first,
                       BidirectionalIterator middle,
                       BidirectionalIterator last, Compare comp);
  template<class ExecutionPolicy, class BidirectionalIterator>
    void inplace_merge(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                       BidirectionalIterator first,
                       BidirectionalIterator middle,
                       BidirectionalIterator last);
  template<class ExecutionPolicy, class BidirectionalIterator, class Compare>
    void inplace_merge(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                       BidirectionalIterator first,
                       BidirectionalIterator middle,
                       BidirectionalIterator last, Compare comp);

  namespace ranges {
    template<BidirectionalIterator I, Sentinel<I> S, class Comp = ranges::less,
             class Proj = identity>
      requires Sortable<I, Comp, Proj>
      I inplace_merge(I first, I middle, S last, Comp comp = {}, Proj proj = {});
    template<BidirectionalRange R, class Comp = ranges::less, class Proj = identity>
      requires Sortable<iterator_t<R>, Comp, Proj>
      safe_iterator_t<R>
        inplace_merge(R&& r, iterator_t<R> middle, Comp comp = {},
                      Proj proj = {});
  }

  // \ref{alg.set.operations}, set operations
  template<class InputIterator1, class InputIterator2>
    constexpr bool includes(InputIterator1 first1, InputIterator1 last1,
                            InputIterator2 first2, InputIterator2 last2);
  template<class InputIterator1, class InputIterator2, class Compare>
    constexpr bool includes(InputIterator1 first1, InputIterator1 last1,
                            InputIterator2 first2, InputIterator2 last2,
                            Compare comp);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2>
    bool includes(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                  ForwardIterator1 first1, ForwardIterator1 last1,
                  ForwardIterator2 first2, ForwardIterator2 last2);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
           class Compare>
    bool includes(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                  ForwardIterator1 first1, ForwardIterator1 last1,
                  ForwardIterator2 first2, ForwardIterator2 last2,
                  Compare comp);

  namespace ranges {
    template<InputIterator I1, Sentinel<I1> S1, InputIterator I2, Sentinel<I2> S2,
             class Proj1 = identity, class Proj2 = identity,
             IndirectStrictWeakOrder<projected<I1, Proj1>, projected<I2, Proj2>> Comp =
               ranges::less>
      constexpr bool includes(I1 first1, S1 last1, I2 first2, S2 last2, Comp comp = {},
                              Proj1 proj1 = {}, Proj2 proj2 = {});
    template<InputRange R1, InputRange R2, class Proj1 = identity,
             class Proj2 = identity,
             IndirectStrictWeakOrder<projected<iterator_t<R1>, Proj1>,
                                     projected<iterator_t<R2>, Proj2>> Comp = ranges::less>
      constexpr bool includes(R1&& r1, R2&& r2, Comp comp = {},
                              Proj1 proj1 = {}, Proj2 proj2 = {});
  }

  template<class InputIterator1, class InputIterator2, class OutputIterator>
    constexpr OutputIterator
      set_union(InputIterator1 first1, InputIterator1 last1,
                InputIterator2 first2, InputIterator2 last2,
                OutputIterator result);
  template<class InputIterator1, class InputIterator2, class OutputIterator, class Compare>
    constexpr OutputIterator
                set_union(InputIterator1 first1, InputIterator1 last1,
                InputIterator2 first2, InputIterator2 last2,
                OutputIterator result, Compare comp);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
           class ForwardIterator>
    ForwardIterator
      set_union(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                ForwardIterator1 first1, ForwardIterator1 last1,
                ForwardIterator2 first2, ForwardIterator2 last2,
                ForwardIterator result);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
           class ForwardIterator, class Compare>
    ForwardIterator
      set_union(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                ForwardIterator1 first1, ForwardIterator1 last1,
                ForwardIterator2 first2, ForwardIterator2 last2,
                ForwardIterator result, Compare comp);

  namespace ranges {
    template<class I1, class I2, class O>
    using set_union_result = binary_transform_result<I1, I2, O>;

    template<InputIterator I1, Sentinel<I1> S1, InputIterator I2, Sentinel<I2> S2,
             WeaklyIncrementable O, class Comp = ranges::less,
             class Proj1 = identity, class Proj2 = identity>
      requires Mergeable<I1, I2, O, Comp, Proj1, Proj2>
      constexpr set_union_result<I1, I2, O>
        set_union(I1 first1, S1 last1, I2 first2, S2 last2, O result, Comp comp = {},
                  Proj1 proj1 = {}, Proj2 proj2 = {});
    template<InputRange R1, InputRange R2, WeaklyIncrementable O,
             class Comp = ranges::less, class Proj1 = identity, class Proj2 = identity>
      requires Mergeable<iterator_t<R1>, iterator_t<R2>, O, Comp, Proj1, Proj2>
      constexpr set_union_result<safe_iterator_t<R1>, safe_iterator_t<R2>, O>
        set_union(R1&& r1, R2&& r2, O result, Comp comp = {},
                  Proj1 proj1 = {}, Proj2 proj2 = {});
  }

  template<class InputIterator1, class InputIterator2, class OutputIterator>
    constexpr OutputIterator
      set_intersection(InputIterator1 first1, InputIterator1 last1,
                       InputIterator2 first2, InputIterator2 last2,
                       OutputIterator result);
  template<class InputIterator1, class InputIterator2, class OutputIterator, class Compare>
    constexpr OutputIterator
      set_intersection(InputIterator1 first1, InputIterator1 last1,
                       InputIterator2 first2, InputIterator2 last2,
                       OutputIterator result, Compare comp);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
           class ForwardIterator>
    ForwardIterator
      set_intersection(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                       ForwardIterator1 first1, ForwardIterator1 last1,
                       ForwardIterator2 first2, ForwardIterator2 last2,
                       ForwardIterator result);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
           class ForwardIterator, class Compare>
    ForwardIterator
      set_intersection(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                       ForwardIterator1 first1, ForwardIterator1 last1,
                       ForwardIterator2 first2, ForwardIterator2 last2,
                       ForwardIterator result, Compare comp);

  namespace ranges {
    template<class I1, class I2, class O>
    using set_intersection_result = binary_transform_result<I1, I2, O>;

    template<InputIterator I1, Sentinel<I1> S1, InputIterator I2, Sentinel<I2> S2,
             WeaklyIncrementable O, class Comp = ranges::less,
             class Proj1 = identity, class Proj2 = identity>
      requires Mergeable<I1, I2, O, Comp, Proj1, Proj2>
      constexpr set_intersection_result<I1, I2, O>
        set_intersection(I1 first1, S1 last1, I2 first2, S2 last2, O result,
                         Comp comp = {}, Proj1 proj1 = {}, Proj2 proj2 = {});
    template<InputRange R1, InputRange R2, WeaklyIncrementable O,
             class Comp = ranges::less, class Proj1 = identity, class Proj2 = identity>
      requires Mergeable<iterator_t<R1>, iterator_t<R2>, O, Comp, Proj1, Proj2>
      constexpr set_intersection_result<safe_iterator_t<R1>, safe_iterator_t<R2>, O>
        set_intersection(R1&& r1, R2&& r2, O result,
                         Comp comp = {}, Proj1 proj1 = {}, Proj2 proj2 = {});
  }

  template<class InputIterator1, class InputIterator2, class OutputIterator>
    constexpr OutputIterator
      set_difference(InputIterator1 first1, InputIterator1 last1,
                     InputIterator2 first2, InputIterator2 last2,
                     OutputIterator result);
  template<class InputIterator1, class InputIterator2, class OutputIterator, class Compare>
    constexpr OutputIterator
      set_difference(InputIterator1 first1, InputIterator1 last1,
                     InputIterator2 first2, InputIterator2 last2,
                     OutputIterator result, Compare comp);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
           class ForwardIterator>
    ForwardIterator
      set_difference(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                     ForwardIterator1 first1, ForwardIterator1 last1,
                     ForwardIterator2 first2, ForwardIterator2 last2,
                     ForwardIterator result);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
           class ForwardIterator, class Compare>
    ForwardIterator
      set_difference(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                     ForwardIterator1 first1, ForwardIterator1 last1,
                     ForwardIterator2 first2, ForwardIterator2 last2,
                     ForwardIterator result, Compare comp);

  namespace ranges {
    template<class I, class O>
    using set_difference_result = copy_result<I, O>;

    template<InputIterator I1, Sentinel<I1> S1, InputIterator I2, Sentinel<I2> S2,
             WeaklyIncrementable O, class Comp = ranges::less,
             class Proj1 = identity, class Proj2 = identity>
      requires Mergeable<I1, I2, O, Comp, Proj1, Proj2>
      constexpr set_difference_result<I1, O>
        set_difference(I1 first1, S1 last1, I2 first2, S2 last2, O result,
                       Comp comp = {}, Proj1 proj1 = {}, Proj2 proj2 = {});
    template<InputRange R1, InputRange R2, WeaklyIncrementable O,
             class Comp = ranges::less, class Proj1 = identity, class Proj2 = identity>
      requires Mergeable<iterator_t<R1>, iterator_t<R2>, O, Comp, Proj1, Proj2>
      constexpr set_difference_result<safe_iterator_t<R1>, O>
        set_difference(R1&& r1, R2&& r2, O result,
                       Comp comp = {}, Proj1 proj1 = {}, Proj2 proj2 = {});
  }

  template<class InputIterator1, class InputIterator2, class OutputIterator>
    constexpr OutputIterator
      set_symmetric_difference(InputIterator1 first1, InputIterator1 last1,
                               InputIterator2 first2, InputIterator2 last2,
                               OutputIterator result);
  template<class InputIterator1, class InputIterator2, class OutputIterator, class Compare>
    constexpr OutputIterator
      set_symmetric_difference(InputIterator1 first1, InputIterator1 last1,
                               InputIterator2 first2, InputIterator2 last2,
                               OutputIterator result, Compare comp);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
           class ForwardIterator>
    ForwardIterator
      set_symmetric_difference(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                               ForwardIterator1 first1, ForwardIterator1 last1,
                               ForwardIterator2 first2, ForwardIterator2 last2,
                               ForwardIterator result);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
           class ForwardIterator, class Compare>
    ForwardIterator
      set_symmetric_difference(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                               ForwardIterator1 first1, ForwardIterator1 last1,
                               ForwardIterator2 first2, ForwardIterator2 last2,
                               ForwardIterator result, Compare comp);

  namespace ranges {
    template<class I1, class I2, class O>
    using set_symmetric_difference_result = binary_transform_result<I1, I2, O>;

    template<InputIterator I1, Sentinel<I1> S1, InputIterator I2, Sentinel<I2> S2,
             WeaklyIncrementable O, class Comp = ranges::less,
             class Proj1 = identity, class Proj2 = identity>
      requires Mergeable<I1, I2, O, Comp, Proj1, Proj2>
      constexpr set_symmetric_difference_result<I1, I2, O>
        set_symmetric_difference(I1 first1, S1 last1, I2 first2, S2 last2, O result,
                                 Comp comp = {}, Proj1 proj1 = {},
                                 Proj2 proj2 = {});
    template<InputRange R1, InputRange R2, WeaklyIncrementable O,
             class Comp = ranges::less, class Proj1 = identity, class Proj2 = identity>
      requires Mergeable<iterator_t<R1>, iterator_t<R2>, O, Comp, Proj1, Proj2>
      constexpr set_symmetric_difference_result<safe_iterator_t<R1>, safe_iterator_t<R2>, O>
        set_symmetric_difference(R1&& r1, R2&& r2, O result, Comp comp = {},
                                 Proj1 proj1 = {}, Proj2 proj2 = {});
  }

  // \ref{alg.heap.operations}, heap operations
  template<class RandomAccessIterator>
    constexpr void push_heap(RandomAccessIterator first, RandomAccessIterator last);
  template<class RandomAccessIterator, class Compare>
    constexpr void push_heap(RandomAccessIterator first, RandomAccessIterator last,
                             Compare comp);

  namespace ranges {
    template<RandomAccessIterator I, Sentinel<I> S, class Comp = ranges::less,
             class Proj = identity>
      requires Sortable<I, Comp, Proj>
      constexpr I
        push_heap(I first, S last, Comp comp = {}, Proj proj = {});
    template<RandomAccessRange R, class Comp = ranges::less, class Proj = identity>
      requires Sortable<iterator_t<R>, Comp, Proj>
      constexpr safe_iterator_t<R>
        push_heap(R&& r, Comp comp = {}, Proj proj = {});
  }

  template<class RandomAccessIterator>
    constexpr void pop_heap(RandomAccessIterator first, RandomAccessIterator last);
  template<class RandomAccessIterator, class Compare>
    constexpr void pop_heap(RandomAccessIterator first, RandomAccessIterator last,
                            Compare comp);

  namespace ranges {
    template<RandomAccessIterator I, Sentinel<I> S, class Comp = ranges::less,
             class Proj = identity>
      requires Sortable<I, Comp, Proj>
      constexpr I
        pop_heap(I first, S last, Comp comp = {}, Proj proj = {});
    template<RandomAccessRange R, class Comp = ranges::less, class Proj = identity>
      requires Sortable<iterator_t<R>, Comp, Proj>
      constexpr safe_iterator_t<R>
        pop_heap(R&& r, Comp comp = {}, Proj proj = {});
  }

  template<class RandomAccessIterator>
    constexpr void make_heap(RandomAccessIterator first, RandomAccessIterator last);
  template<class RandomAccessIterator, class Compare>
    constexpr void make_heap(RandomAccessIterator first, RandomAccessIterator last,
                             Compare comp);

  namespace ranges {
    template<RandomAccessIterator I, Sentinel<I> S, class Comp = ranges::less,
             class Proj = identity>
      requires Sortable<I, Comp, Proj>
      constexpr I
        make_heap(I first, S last, Comp comp = {}, Proj proj = {});
    template<RandomAccessRange R, class Comp = ranges::less, class Proj = identity>
      requires Sortable<iterator_t<R>, Comp, Proj>
      constexpr safe_iterator_t<R>
        make_heap(R&& r, Comp comp = {}, Proj proj = {});
  }

  template<class RandomAccessIterator>
    constexpr void sort_heap(RandomAccessIterator first, RandomAccessIterator last);
  template<class RandomAccessIterator, class Compare>
    constexpr void sort_heap(RandomAccessIterator first, RandomAccessIterator last,
                             Compare comp);

  namespace ranges {
    template<RandomAccessIterator I, Sentinel<I> S, class Comp = ranges::less,
             class Proj = identity>
      requires Sortable<I, Comp, Proj>
      constexpr I
        sort_heap(I first, S last, Comp comp = {}, Proj proj = {});
    template<RandomAccessRange R, class Comp = ranges::less, class Proj = identity>
      requires Sortable<iterator_t<R>, Comp, Proj>
      constexpr safe_iterator_t<R>
        sort_heap(R&& r, Comp comp = {}, Proj proj = {});
  }

  template<class RandomAccessIterator>
    constexpr bool is_heap(RandomAccessIterator first, RandomAccessIterator last);
  template<class RandomAccessIterator, class Compare>
    constexpr bool is_heap(RandomAccessIterator first, RandomAccessIterator last,
                           Compare comp);
  template<class ExecutionPolicy, class RandomAccessIterator>
    bool is_heap(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                 RandomAccessIterator first, RandomAccessIterator last);
  template<class ExecutionPolicy, class RandomAccessIterator, class Compare>
    bool is_heap(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                 RandomAccessIterator first, RandomAccessIterator last,
                 Compare comp);

  namespace ranges {
    template<RandomAccessIterator I, Sentinel<I> S, class Proj = identity,
             IndirectStrictWeakOrder<projected<I, Proj>> Comp = ranges::less>
      constexpr bool is_heap(I first, S last, Comp comp = {}, Proj proj = {});
    template<RandomAccessRange R, class Proj = identity,
             IndirectStrictWeakOrder<projected<iterator_t<R>, Proj>> Comp = ranges::less>
      constexpr bool is_heap(R&& r, Comp comp = {}, Proj proj = {});
  }

  template<class RandomAccessIterator>
    constexpr RandomAccessIterator
      is_heap_until(RandomAccessIterator first, RandomAccessIterator last);
  template<class RandomAccessIterator, class Compare>
    constexpr RandomAccessIterator
      is_heap_until(RandomAccessIterator first, RandomAccessIterator last,
                    Compare comp);
  template<class ExecutionPolicy, class RandomAccessIterator>
    RandomAccessIterator
      is_heap_until(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                    RandomAccessIterator first, RandomAccessIterator last);
  template<class ExecutionPolicy, class RandomAccessIterator, class Compare>
    RandomAccessIterator
      is_heap_until(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                    RandomAccessIterator first, RandomAccessIterator last,
                    Compare comp);

  namespace ranges {
    template<RandomAccessIterator I, Sentinel<I> S, class Proj = identity,
             IndirectStrictWeakOrder<projected<I, Proj>> Comp = ranges::less>
      constexpr I is_heap_until(I first, S last, Comp comp = {}, Proj proj = {});
    template<RandomAccessRange R, class Proj = identity,
             IndirectStrictWeakOrder<projected<iterator_t<R>, Proj>> Comp = ranges::less>
      constexpr safe_iterator_t<R>
        is_heap_until(R&& r, Comp comp = {}, Proj proj = {});
  }

  // \ref{alg.min.max}, minimum and maximum
  template<class T> constexpr const T& min(const T& a, const T& b);
  template<class T, class Compare>
    constexpr const T& min(const T& a, const T& b, Compare comp);
  template<class T>
    constexpr T min(initializer_list<T> t);
  template<class T, class Compare>
    constexpr T min(initializer_list<T> t, Compare comp);

  namespace ranges {
    template<class T, class Proj = identity,
             IndirectStrictWeakOrder<projected<const T*, Proj>> Comp = ranges::less>
      constexpr const T& min(const T& a, const T& b, Comp comp = {}, Proj proj = {});
    template<Copyable T, class Proj = identity,
             IndirectStrictWeakOrder<projected<const T*, Proj>> Comp = ranges::less>
      constexpr T min(initializer_list<T> r, Comp comp = {}, Proj proj = {});
    template<InputRange R, class Proj = identity,
             IndirectStrictWeakOrder<projected<iterator_t<R>, Proj>> Comp = ranges::less>
      requires IndirectlyCopyableStorable<iterator_t<R>, iter_value_t<iterator_t<R>>*>
      constexpr iter_value_t<iterator_t<R>>
        min(R&& r, Comp comp = {}, Proj proj = {});
  }

  template<class T> constexpr const T& max(const T& a, const T& b);
  template<class T, class Compare>
    constexpr const T& max(const T& a, const T& b, Compare comp);
  template<class T>
    constexpr T max(initializer_list<T> t);
  template<class T, class Compare>
    constexpr T max(initializer_list<T> t, Compare comp);

  namespace ranges {
    template<class T, class Proj = identity,
             IndirectStrictWeakOrder<projected<const T*, Proj>> Comp = ranges::less>
      constexpr const T& max(const T& a, const T& b, Comp comp = {}, Proj proj = {});
    template<Copyable T, class Proj = identity,
             IndirectStrictWeakOrder<projected<const T*, Proj>> Comp = ranges::less>
      constexpr T max(initializer_list<T> r, Comp comp = {}, Proj proj = {});
    template<InputRange R, class Proj = identity,
             IndirectStrictWeakOrder<projected<iterator_t<R>, Proj>> Comp = ranges::less>
      requires IndirectlyCopyableStorable<iterator_t<R>, iter_value_t<iterator_t<R>>*>
      constexpr iter_value_t<iterator_t<R>>
        max(R&& r, Comp comp = {}, Proj proj = {});
  }

  template<class T> constexpr pair<const T&, const T&> minmax(const T& a, const T& b);
  template<class T, class Compare>
    constexpr pair<const T&, const T&> minmax(const T& a, const T& b, Compare comp);
  template<class T>
    constexpr pair<T, T> minmax(initializer_list<T> t);
  template<class T, class Compare>
    constexpr pair<T, T> minmax(initializer_list<T> t, Compare comp);

  namespace ranges {
    template<class T>
    struct minmax_result {
      [[no_unique_address]] T min;
      [[no_unique_address]] T max;

      template<class T2>
        requires ConvertibleTo<const T&, T2>
        operator minmax_result<T2>() const & {
          return {min, max};
        }

      template<class T2>
        requires ConvertibleTo<T, T2>
        operator minmax_result<T2>() && {
          return {std::move(min), std::move(max)};
        }
    };

    template<class T, class Proj = identity,
             IndirectStrictWeakOrder<projected<const T*, Proj>> Comp = ranges::less>
      constexpr minmax_result<const T&>
        minmax(const T& a, const T& b, Comp comp = {}, Proj proj = {});
    template<Copyable T, class Proj = identity,
             IndirectStrictWeakOrder<projected<const T*, Proj>> Comp = ranges::less>
      constexpr minmax_result<T>
        minmax(initializer_list<T> r, Comp comp = {}, Proj proj = {});
    template<InputRange R, class Proj = identity,
             IndirectStrictWeakOrder<projected<iterator_t<R>, Proj>> Comp = ranges::less>
      requires IndirectlyCopyableStorable<iterator_t<R>, iter_value_t<iterator_t<R>>*>
      constexpr minmax_result<iter_value_t<iterator_t<R>>>
        minmax(R&& r, Comp comp = {}, Proj proj = {});
  }

  template<class ForwardIterator>
    constexpr ForwardIterator min_element(ForwardIterator first, ForwardIterator last);
  template<class ForwardIterator, class Compare>
    constexpr ForwardIterator min_element(ForwardIterator first, ForwardIterator last,
                                          Compare comp);
  template<class ExecutionPolicy, class ForwardIterator>
    ForwardIterator min_element(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                                ForwardIterator first, ForwardIterator last);
  template<class ExecutionPolicy, class ForwardIterator, class Compare>
    ForwardIterator min_element(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                                ForwardIterator first, ForwardIterator last,
                                Compare comp);

  namespace ranges {
    template<ForwardIterator I, Sentinel<I> S, class Proj = identity,
             IndirectStrictWeakOrder<projected<I, Proj>> Comp = ranges::less>
      constexpr I min_element(I first, S last, Comp comp = {}, Proj proj = {});
    template<ForwardRange R, class Proj = identity,
             IndirectStrictWeakOrder<projected<iterator_t<R>, Proj>> Comp = ranges::less>
      constexpr safe_iterator_t<R>
        min_element(R&& r, Comp comp = {}, Proj proj = {});
  }

  template<class ForwardIterator>
    constexpr ForwardIterator max_element(ForwardIterator first, ForwardIterator last);
  template<class ForwardIterator, class Compare>
    constexpr ForwardIterator max_element(ForwardIterator first, ForwardIterator last,
                                          Compare comp);
  template<class ExecutionPolicy, class ForwardIterator>
    ForwardIterator max_element(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                                ForwardIterator first, ForwardIterator last);
  template<class ExecutionPolicy, class ForwardIterator, class Compare>
    ForwardIterator max_element(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                                ForwardIterator first, ForwardIterator last,
                                Compare comp);

 namespace ranges {
    template<ForwardIterator I, Sentinel<I> S, class Proj = identity,
             IndirectStrictWeakOrder<projected<I, Proj>> Comp = ranges::less>
      constexpr I max_element(I first, S last, Comp comp = {}, Proj proj = {});
    template<ForwardRange R, class Proj = identity,
             IndirectStrictWeakOrder<projected<iterator_t<R>, Proj>> Comp = ranges::less>
      constexpr safe_iterator_t<R>
        max_element(R&& r, Comp comp = {}, Proj proj = {});
  }

  template<class ForwardIterator>
    constexpr pair<ForwardIterator, ForwardIterator>
      minmax_element(ForwardIterator first, ForwardIterator last);
  template<class ForwardIterator, class Compare>
    constexpr pair<ForwardIterator, ForwardIterator>
      minmax_element(ForwardIterator first, ForwardIterator last, Compare comp);
  template<class ExecutionPolicy, class ForwardIterator>
    pair<ForwardIterator, ForwardIterator>
      minmax_element(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                     ForwardIterator first, ForwardIterator last);
  template<class ExecutionPolicy, class ForwardIterator, class Compare>
    pair<ForwardIterator, ForwardIterator>
      minmax_element(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                     ForwardIterator first, ForwardIterator last, Compare comp);

  namespace ranges {
    template<class I>
    using minmax_element_result = minmax_result<I>;

    template<ForwardIterator I, Sentinel<I> S, class Proj = identity,
             IndirectStrictWeakOrder<projected<I, Proj>> Comp = ranges::less>
      constexpr minmax_element_result<I>
        minmax_element(I first, S last, Comp comp = {}, Proj proj = {});
    template<ForwardRange R, class Proj = identity,
             IndirectStrictWeakOrder<projected<iterator_t<R>, Proj>> Comp = ranges::less>
      constexpr minmax_element_result<safe_iterator_t<R>>
        minmax_element(R&& r, Comp comp = {}, Proj proj = {});
  }

  // \ref{alg.clamp}, bounded value
  template<class T>
    constexpr const T& clamp(const T& v, const T& lo, const T& hi);
  template<class T, class Compare>
    constexpr const T& clamp(const T& v, const T& lo, const T& hi, Compare comp);

  // \ref{alg.lex.comparison}, lexicographical comparison
  template<class InputIterator1, class InputIterator2>
    constexpr bool
      lexicographical_compare(InputIterator1 first1, InputIterator1 last1,
                              InputIterator2 first2, InputIterator2 last2);
  template<class InputIterator1, class InputIterator2, class Compare>
    constexpr bool
      lexicographical_compare(InputIterator1 first1, InputIterator1 last1,
                              InputIterator2 first2, InputIterator2 last2,
                              Compare comp);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2>
    bool
      lexicographical_compare(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                              ForwardIterator1 first1, ForwardIterator1 last1,
                              ForwardIterator2 first2, ForwardIterator2 last2);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
           class Compare>
    bool
      lexicographical_compare(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                              ForwardIterator1 first1, ForwardIterator1 last1,
                              ForwardIterator2 first2, ForwardIterator2 last2,
                              Compare comp);

  namespace ranges {
    template<InputIterator I1, Sentinel<I1> S1, InputIterator I2, Sentinel<I2> S2,
             class Proj1 = identity, class Proj2 = identity,
             IndirectStrictWeakOrder<projected<I1, Proj1>, projected<I2, Proj2>> Comp =
               ranges::less>
      constexpr bool
        lexicographical_compare(I1 first1, S1 last1, I2 first2, S2 last2,
                                Comp comp = {}, Proj1 proj1 = {}, Proj2 proj2 = {});
    template<InputRange R1, InputRange R2, class Proj1 = identity,
             class Proj2 = identity,
             IndirectStrictWeakOrder<projected<iterator_t<R1>, Proj1>,
                                     projected<iterator_t<R2>, Proj2>> Comp = ranges::less>
      constexpr bool
        lexicographical_compare(R1&& r1, R2&& r2, Comp comp = {},
                                Proj1 proj1 = {}, Proj2 proj2 = {});
  }

  // \ref{alg.3way}, three-way comparison algorithms
  template<class T, class U>
    constexpr auto compare_3way(const T& a, const U& b);
  template<class InputIterator1, class InputIterator2, class Cmp>
    constexpr auto
      lexicographical_compare_3way(InputIterator1 b1, InputIterator1 e1,
                                   InputIterator2 b2, InputIterator2 e2,
                                   Cmp comp)
        -> common_comparison_category_t<decltype(comp(*b1, *b2)), strong_ordering>;
  template<class InputIterator1, class InputIterator2>
    constexpr auto
      lexicographical_compare_3way(InputIterator1 b1, InputIterator1 e1,
                                   InputIterator2 b2, InputIterator2 e2);

  // \ref{alg.permutation.generators}, permutations
  template<class BidirectionalIterator>
    constexpr bool next_permutation(BidirectionalIterator first,
                                    BidirectionalIterator last);
  template<class BidirectionalIterator, class Compare>
    constexpr bool next_permutation(BidirectionalIterator first,
                                    BidirectionalIterator last, Compare comp);

  namespace ranges {
    template<BidirectionalIterator I, Sentinel<I> S, class Comp = ranges::less,
             class Proj = identity>
      requires Sortable<I, Comp, Proj>
      constexpr bool
        next_permutation(I first, S last, Comp comp = {}, Proj proj = {});
    template<BidirectionalRange R, class Comp = ranges::less,
             class Proj = identity>
      requires Sortable<iterator_t<R>, Comp, Proj>
      constexpr bool
        next_permutation(R&& r, Comp comp = {}, Proj proj = {});
  }

  template<class BidirectionalIterator>
    constexpr bool prev_permutation(BidirectionalIterator first,
                                    BidirectionalIterator last);
  template<class BidirectionalIterator, class Compare>
    constexpr bool prev_permutation(BidirectionalIterator first,
                                    BidirectionalIterator last, Compare comp);

  namespace ranges {
    template<BidirectionalIterator I, Sentinel<I> S, class Comp = ranges::less,
             class Proj = identity>
      requires Sortable<I, Comp, Proj>
      constexpr bool
        prev_permutation(I first, S last, Comp comp = {}, Proj proj = {});
    template<BidirectionalRange R, class Comp = ranges::less,
             class Proj = identity>
      requires Sortable<iterator_t<R>, Comp, Proj>
      constexpr bool
        prev_permutation(R&& r, Comp comp = {}, Proj proj = {});
  }
}
\end{codeblock}

\rSec1[alg.nonmodifying]{Non-modifying sequence operations}

\rSec2[alg.all.of]{All of}

\indexlibrary{\idxcode{all_of}}%
\begin{itemdecl}
template<class InputIterator, class Predicate>
  constexpr bool all_of(InputIterator first, InputIterator last, Predicate pred);
template<class ExecutionPolicy, class ForwardIterator, class Predicate>
  bool all_of(ExecutionPolicy&& exec, ForwardIterator first, ForwardIterator last,
              Predicate pred);

template<InputIterator I, Sentinel<I> S, class Proj = identity,
         IndirectUnaryPredicate<projected<I, Proj>> Pred>
  constexpr bool ranges::all_of(I first, S last, Pred pred, Proj proj = {});
template<InputRange R, class Proj = identity,
         IndirectUnaryPredicate<projected<iterator_t<R>, Proj>> Pred>
  constexpr bool ranges::all_of(R&& r, Pred pred, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let $E$ be \tcode{pred(*i)} and \tcode{invoke(pred, invoke(proj, *i))}
for the overloads in namespace \tcode{std} and \tcode{std::ranges}, respectively.

\pnum
\returns
\tcode{false} if $E$ is \tcode{false}
for some iterator \tcode{i} in the range \range{first}{last}, and
\tcode{true} otherwise.

\pnum
\complexity
At most \tcode{last - first} applications of the predicate and any projection.
\end{itemdescr}

\rSec2[alg.any.of]{Any of}

\indexlibrary{\idxcode{any_of}}%
\begin{itemdecl}
template<class InputIterator, class Predicate>
  constexpr bool any_of(InputIterator first, InputIterator last, Predicate pred);
template<class ExecutionPolicy, class ForwardIterator, class Predicate>
  bool any_of(ExecutionPolicy&& exec, ForwardIterator first, ForwardIterator last,
              Predicate pred);

template<InputIterator I, Sentinel<I> S, class Proj = identity,
         IndirectUnaryPredicate<projected<I, Proj>> Pred>
  constexpr bool ranges::any_of(I first, S last, Pred pred, Proj proj = {});
template<InputRange R, class Proj = identity,
         IndirectUnaryPredicate<projected<iterator_t<R>, Proj>> Pred>
  constexpr bool ranges::any_of(R&& r, Pred pred, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let $E$ be \tcode{pred(*i)} and \tcode{invoke(pred, invoke(proj, *i))}
for the overloads in namespace \tcode{std} and \tcode{std::ranges}, respectively.

\pnum
\returns
\tcode{true} if $E$ is \tcode{true} for some iterator \tcode{i}
in the range \range{first}{last}, and \tcode{false} otherwise.

\pnum
\complexity At most \tcode{last - first} applications of the predicate
and any projection.
\end{itemdescr}

\rSec2[alg.none.of]{None of}

\indexlibrary{\idxcode{none_of}}%
\begin{itemdecl}
template<class InputIterator, class Predicate>
  constexpr bool none_of(InputIterator first, InputIterator last, Predicate pred);
template<class ExecutionPolicy, class ForwardIterator, class Predicate>
  bool none_of(ExecutionPolicy&& exec, ForwardIterator first, ForwardIterator last,
               Predicate pred);

template<InputIterator I, Sentinel<I> S, class Proj = identity,
         IndirectUnaryPredicate<projected<I, Proj>> Pred>
  constexpr bool ranges::none_of(I first, S last, Pred pred, Proj proj = {});
template<InputRange R, class Proj = identity,
         IndirectUnaryPredicate<projected<iterator_t<R>, Proj>> Pred>
  constexpr bool ranges::none_of(R&& r, Pred pred, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let $E$ be \tcode{pred(*i)} and \tcode{invoke(pred, invoke(proj, *i))}
for the overloads in namespace \tcode{std} and \tcode{std::ranges}, respectively.

\pnum
\returns
\tcode{false} if $E$ is \tcode{true}
for some iterator \tcode{i} in the range \range{first}{last}, and
\tcode{true} otherwise.

\pnum
\complexity
At most \tcode{last - first} applications of the predicate and any projection.
\end{itemdescr}

\rSec2[alg.foreach]{For each}

\indexlibrary{\idxcode{for_each}}%
\begin{itemdecl}
template<class InputIterator, class Function>
  constexpr Function for_each(InputIterator first, InputIterator last, Function f);
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
\tcode{Function} shall satisfy
the \oldconcept{MoveConstructible} requirements (\tref{cpp17.moveconstructible}).
\begin{note}
\tcode{Function} need not meet the requirements of
\oldconcept{CopyConstructible} (\tref{cpp17.copyconstructible}).
\end{note}

\pnum
\effects
Applies \tcode{f} to the result of dereferencing
every iterator in the range \range{first}{last},
starting from \tcode{first} and proceeding to \tcode{last - 1}.
\begin{note}
If the type of \tcode{first} satisfies the requirements of a mutable iterator,
\tcode{f} may apply non-constant functions through the dereferenced iterator.
\end{note}

\pnum
\returns
\tcode{f}.

\pnum
\complexity
Applies \tcode{f} exactly \tcode{last - first} times.

\pnum
\remarks
If \tcode{f} returns a result, the result is ignored.
\end{itemdescr}

\indexlibrary{\idxcode{for_each}}%
\begin{itemdecl}
template<class ExecutionPolicy, class ForwardIterator, class Function>
  void for_each(ExecutionPolicy&& exec,
                ForwardIterator first, ForwardIterator last,
                Function f);
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
\tcode{Function} shall satisfy the \oldconcept{CopyConstructible} requirements.

\pnum
\effects
Applies \tcode{f} to the result of dereferencing
every iterator in the range \range{first}{last}.
\begin{note}
If the type of \tcode{first} satisfies the requirements of a mutable iterator,
\tcode{f} may apply non-constant functions through the dereferenced iterator.
\end{note}

\pnum
\complexity
Applies \tcode{f} exactly \tcode{last - first} times.

\pnum
\remarks
If \tcode{f} returns a result, the result is ignored.
Implementations do not have
the freedom granted under \ref{algorithms.parallel.exec}
to make arbitrary copies of elements from the input sequence.

\pnum
\begin{note}
Does not return a copy of its \tcode{Function} parameter,
since parallelization may not permit efficient state accumulation.
\end{note}
\end{itemdescr}

\indexlibrary{\idxcode{for_each}}%
\begin{itemdecl}
template<InputIterator I, Sentinel<I> S, class Proj = identity,
         IndirectUnaryInvocable<projected<I, Proj>> Fun>
  constexpr ranges::for_each_result<I, Fun>
    ranges::for_each(I first, S last, Fun f, Proj proj = {});
template<InputRange R, class Proj = identity,
         IndirectUnaryInvocable<projected<iterator_t<R>, Proj>> Fun>
  constexpr ranges::for_each_result<safe_iterator_t<R>, Fun>
    ranges::for_each(R&& r, Fun f, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
\effects
Calls \tcode{invoke(f, invoke(proj, *i))}
for every iterator \tcode{i} in the range \range{first}{last},
starting from \tcode{first} and proceeding to \tcode{last - 1}.
\begin{note}
If the result of \tcode{invoke(proj, *i)} is a mutable reference,
\tcode{f} may apply non-constant functions.
\end{note}

\pnum
\returns
\tcode{\{last, std::move(f)\}}.

\pnum
\complexity
Applies \tcode{f} and \tcode{proj} exactly \tcode{last - first} times.

\pnum
\remarks
If \tcode{f} returns a result, the result is ignored.

\pnum
\begin{note}
The overloads in namespace \tcode{ranges} require
\tcode{Fun} to model \libconcept{CopyConstructible}.
\end{note}
\end{itemdescr}

\indexlibrary{\idxcode{for_each_n}}%
\begin{itemdecl}
template<class InputIterator, class Size, class Function>
  constexpr InputIterator for_each_n(InputIterator first, Size n, Function f);
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
\tcode{Function} shall satisfy the \oldconcept{MoveConstructible} requirements
\begin{note}
\tcode{Function} need not meet
the requirements of \oldconcept{CopyConstructible}.
\end{note}

\pnum
\requires
\tcode{n >= 0}.

\pnum
\effects
Applies \tcode{f} to the result of dereferencing
every iterator in the range \range{first}{first + n} in order.
\begin{note}
If the type of \tcode{first} satisfies the requirements of a mutable iterator,
\tcode{f} may apply non-constant functions through the dereferenced iterator.
\end{note}

\pnum
\returns
\tcode{first + n}.

\pnum
\remarks
If \tcode{f} returns a result, the result is ignored.
\end{itemdescr}

\indexlibrary{\idxcode{for_each_n}}%
\begin{itemdecl}
template<class ExecutionPolicy, class ForwardIterator, class Size, class Function>
  ForwardIterator for_each_n(ExecutionPolicy&& exec, ForwardIterator first, Size n,
                             Function f);
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
\tcode{Function} shall satisfy the \oldconcept{CopyConstructible} requirements.

\pnum
\requires
\tcode{n >= 0}.

\pnum
\effects
Applies \tcode{f} to the result of dereferencing
every iterator in the range \range{first}{first + n}.
\begin{note}
If the type of \tcode{first} satisfies the requirements of a mutable iterator,
\tcode{f} may apply non-constant functions through the dereferenced iterator.
\end{note}

\pnum
\returns
\tcode{first + n}.

\pnum
\remarks
If \tcode{f} returns a result, the result is ignored.
Implementations do not have
the freedom granted under \ref{algorithms.parallel.exec}
to make arbitrary copies of elements from the input sequence.
\end{itemdescr}

\rSec2[alg.find]{Find}

\indexlibrary{\idxcode{find}}%
\indexlibrary{\idxcode{find_if}}%
\indexlibrary{\idxcode{find_if_not}}%
\begin{itemdecl}
template<class InputIterator, class T>
  constexpr InputIterator find(InputIterator first, InputIterator last,
                               const T& value);
template<class ExecutionPolicy, class ForwardIterator, class T>
  ForwardIterator find(ExecutionPolicy&& exec, ForwardIterator first, ForwardIterator last,
                       const T& value);

template<class InputIterator, class Predicate>
  constexpr InputIterator find_if(InputIterator first, InputIterator last,
                                  Predicate pred);
template<class ExecutionPolicy, class ForwardIterator, class Predicate>
  ForwardIterator find_if(ExecutionPolicy&& exec, ForwardIterator first, ForwardIterator last,
                          Predicate pred);

template<class InputIterator, class Predicate>
  constexpr InputIterator find_if_not(InputIterator first, InputIterator last,
                                      Predicate pred);
template<class ExecutionPolicy, class ForwardIterator, class Predicate>
  ForwardIterator find_if_not(ExecutionPolicy&& exec,
                              ForwardIterator first, ForwardIterator last,
                              Predicate pred);

template<InputIterator I, Sentinel<I> S, class T, class Proj = identity>
  requires IndirectRelation<ranges::equal_to, projected<I, Proj>, const T*>
  constexpr I ranges::find(I first, S last, const T& value, Proj proj = {});
template<InputRange R, class T, class Proj = identity>
  requires IndirectRelation<ranges::equal_to, projected<iterator_t<R>, Proj>, const T*>
  constexpr safe_iterator_t<R>
    ranges::find(R&& r, const T& value, Proj proj = {});
template<InputIterator I, Sentinel<I> S, class Proj = identity,
         IndirectUnaryPredicate<projected<I, Proj>> Pred>
  constexpr I ranges::find_if(I first, S last, Pred pred, Proj proj = {});
template<InputRange R, class Proj = identity,
         IndirectUnaryPredicate<projected<iterator_t<R>, Proj>> Pred>
  constexpr safe_iterator_t<R>
    ranges::find_if(R&& r, Pred pred, Proj proj = {});
template<InputIterator I, Sentinel<I> S, class Proj = identity,
         IndirectUnaryPredicate<projected<I, Proj>> Pred>
  constexpr I ranges::find_if_not(I first, S last, Pred pred, Proj proj = {});
template<InputRange R, class Proj = identity,
         IndirectUnaryPredicate<projected<iterator_t<R>, Proj>> Pred>
  constexpr safe_iterator_t<R>
    ranges::find_if_not(R&& r, Pred pred, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let $E$ be:
\begin{itemize}
\item \tcode{*i == value} for \tcode{find},
\item \tcode{pred(*i) != false} for \tcode{find_if},
\item \tcode{pred(*i) == false} for \tcode{find_if_not},
\item \tcode{invoke(proj, *i) == value} for \tcode{ranges::find},
\item \tcode{invoke(pred, invoke(proj, *i)) != false} for \tcode{ranges::find_if},
\item \tcode{invoke(pred, invoke(proj, *i)) == false} for \tcode{ranges::find_if_not}.
\end{itemize}

\pnum
\returns
The first iterator \tcode{i} in the range \range{first}{last}
for which $E$ is \tcode{true}.
Returns \tcode{last} if no such iterator is found.

\pnum
\complexity
At most \tcode{last - first} applications
of the corresponding predicate and any projection.
\end{itemdescr}

\rSec2[alg.find.end]{Find end}

\indexlibrary{\idxcode{find_end}}%
\begin{itemdecl}
template<class ForwardIterator1, class ForwardIterator2>
  constexpr ForwardIterator1
    find_end(ForwardIterator1 first1, ForwardIterator1 last1,
             ForwardIterator2 first2, ForwardIterator2 last2);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2>
  ForwardIterator1
    find_end(ExecutionPolicy&& exec,
             ForwardIterator1 first1, ForwardIterator1 last1,
             ForwardIterator2 first2, ForwardIterator2 last2);

template<class ForwardIterator1, class ForwardIterator2,
         class BinaryPredicate>
  constexpr ForwardIterator1
    find_end(ForwardIterator1 first1, ForwardIterator1 last1,
             ForwardIterator2 first2, ForwardIterator2 last2,
             BinaryPredicate pred);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
         class BinaryPredicate>
  ForwardIterator1
    find_end(ExecutionPolicy&& exec,
             ForwardIterator1 first1, ForwardIterator1 last1,
             ForwardIterator2 first2, ForwardIterator2 last2,
             BinaryPredicate pred);

template<ForwardIterator I1, Sentinel<I1> S1, ForwardIterator I2, Sentinel<I2> S2,
         class Pred = ranges::equal_to, class Proj1 = identity, class Proj2 = identity>
  requires IndirectlyComparable<I1, I2, Pred, Proj1, Proj2>
  constexpr subrange<I1>
    ranges::find_end(I1 first1, S1 last1, I2 first2, S2 last2, Pred pred = {},
             Proj1 proj1 = {}, Proj2 proj2 = {});
template<ForwardRange R1, ForwardRange R2,
         class Pred = ranges::equal_to, class Proj1 = identity, class Proj2 = identity>
  requires IndirectlyComparable<iterator_t<R1>, iterator_t<R2>, Pred, Proj1, Proj2>
  constexpr safe_subrange_t<R1>
    ranges::find_end(R1&& r1, R2&& r2, Pred pred = {},
             Proj1 proj1 = {}, Proj2 proj2 = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let:
\begin{itemize}
\item
  \tcode{pred} be \tcode{equal_to\{\}}
  for the overloads with no parameter \tcode{pred}.
\item
  $E$ be:
  \begin{itemize}
  \item
    \tcode{pred(*(i + n), *(first2 + n))}
    for the overloads in namespace \tcode{std},
  \item
    \tcode{invoke(pred, invoke(proj1, *(i + n)), invoke(proj2, *(first2 + n)))}
    for the overloads in namespace \tcode{ranges}.
  \end{itemize}
\item
  \tcode{i} be \tcode{last1} if \range{first2}{last2} is empty,
  or if \tcode{(last2 - first2) > (last1 - first1)} is \tcode{true},
  or if there is no iterator
  in the range \range{first1}{last1 - (last2 - first2)}
  such that for every non-negative integer
  \tcode{n < (last2 - first2)}, $E$ is \tcode{true}.
  Otherwise \tcode{i} is the last such iterator
  in \range{first1}{last1 - (last2 - first2)}.
\end{itemize}

\pnum
\returns
\begin{itemize}
\item \tcode{i} for the overloads in namespace \tcode{std}, and
\item \tcode{\{i, i + (i == last1 ? 0 : last2 - first2)\}}
  for the overloads in namespace \tcode{ranges}.
\end{itemize}

\pnum
\complexity
At most
\tcode{(last2 - first2) * (last1 - first1 - (last2 - first2) + 1)}
applications of the corresponding predicate and any projections.
\end{itemdescr}

\rSec2[alg.find.first.of]{Find first}

\indexlibrary{\idxcode{find_first_of}}%
\begin{itemdecl}
template<class InputIterator, class ForwardIterator>
  constexpr InputIterator
    find_first_of(InputIterator first1, InputIterator last1,
                  ForwardIterator first2, ForwardIterator last2);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2>
  ForwardIterator1
    find_first_of(ExecutionPolicy&& exec,
                  ForwardIterator1 first1, ForwardIterator1 last1,
                  ForwardIterator2 first2, ForwardIterator2 last2);

template<class InputIterator, class ForwardIterator,
         class BinaryPredicate>
  constexpr InputIterator
    find_first_of(InputIterator first1, InputIterator last1,
                  ForwardIterator first2, ForwardIterator last2,
                  BinaryPredicate pred);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
         class BinaryPredicate>
  ForwardIterator1
    find_first_of(ExecutionPolicy&& exec,
                  ForwardIterator1 first1, ForwardIterator1 last1,
                  ForwardIterator2 first2, ForwardIterator2 last2,
                  BinaryPredicate pred);

template<InputIterator I1, Sentinel<I1> S1, ForwardIterator I2, Sentinel<I2> S2,
         class Proj1 = identity, class Proj2 = identity,
         IndirectRelation<projected<I1, Proj1>,
                          projected<I2, Proj2>> Pred = ranges::equal_to>
  constexpr I1 ranges::find_first_of(I1 first1, S1 last1, I2 first2, S2 last2,
                                     Pred pred = {},
                                     Proj1 proj1 = {}, Proj2 proj2 = {});
template<InputRange R1, ForwardRange R2, class Proj1 = identity,
         class Proj2 = identity,
         IndirectRelation<projected<iterator_t<R1>, Proj1>,
                          projected<iterator_t<R2>, Proj2>> Pred = ranges::equal_to>
  constexpr safe_iterator_t<R1>
    ranges::find_first_of(R1&& r1, R2&& r2,
                          Pred pred = {},
                          Proj1 proj1 = {}, Proj2 proj2 = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let $E$ be:
\begin{itemize}
\item \tcode{*i == *j} for the overloads with no parameter \tcode{pred},
\item \tcode{pred(*i, *j) != false} for the overloads with a parameter \tcode{pred} and no parameter \tcode{proj1},
\item \tcode{invoke(pred, invoke(proj1, *i), invoke(proj2, *j)) != false} for the overloads with parameters \tcode{pred} and \tcode{proj1}.
\end{itemize}

\pnum
\effects
Finds an element that matches one of a set of values.

\pnum
\returns
The first iterator \tcode{i} in the range \range{first1}{last1}
such that for some iterator \tcode{j} in the range \range{first2}{last2}
$E$ holds.
Returns \tcode{last1}
if \range{first2}{last2} is empty or
if no such iterator is found.

\pnum
\complexity
At most \tcode{(last1-first1) * (last2-first2)} applications
of the corresponding predicate and any projections.
\end{itemdescr}

\rSec2[alg.adjacent.find]{Adjacent find}

\indexlibrary{\idxcode{adjacent_find}}%
\begin{itemdecl}
template<class ForwardIterator>
  constexpr ForwardIterator
    adjacent_find(ForwardIterator first, ForwardIterator last);
template<class ExecutionPolicy, class ForwardIterator>
  ForwardIterator
    adjacent_find(ExecutionPolicy&& exec,
                  ForwardIterator first, ForwardIterator last);

template<class ForwardIterator, class BinaryPredicate>
  constexpr ForwardIterator
    adjacent_find(ForwardIterator first, ForwardIterator last,
                  BinaryPredicate pred);
template<class ExecutionPolicy, class ForwardIterator, class BinaryPredicate>
  ForwardIterator
    adjacent_find(ExecutionPolicy&& exec,
                  ForwardIterator first, ForwardIterator last,
                  BinaryPredicate pred);

template<ForwardIterator I, Sentinel<I> S, class Proj = identity,
         IndirectRelation<projected<I, Proj>> Pred = ranges::equal_to>
  constexpr I ranges::adjacent_find(I first, S last, Pred pred = {}, Proj proj = {});
template<ForwardRange R, class Proj = identity,
         IndirectRelation<projected<iterator_t<R>, Proj>> Pred = ranges::equal_to>
  constexpr safe_iterator_t<R> ranges::adjacent_find(R&& r, Pred pred = {}, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let $E$ be:
\begin{itemize}
\setlength{\emergencystretch}{1em}
\item \tcode{*i == *(i + 1)} for the overloads with no parameter \tcode{pred},
\item \tcode{pred(*i, *(i + 1)) != false} for the overloads with a parameter \tcode{pred} and no parameter \tcode{proj},
\item \tcode{invoke(pred, invoke(proj, *i), invoke(proj, *(i + 1))) != false} for the overloads with both parameters \tcode{pred} and \tcode{proj}.
\end{itemize}

\pnum
\returns
The first iterator \tcode{i}
such that both \tcode{i} and \tcode{i + 1} are in the range \range{first}{last}
for which $E$ holds.
Returns \tcode{last} if no such iterator is found.

\pnum
\complexity
For the overloads with no \tcode{ExecutionPolicy},
exactly \[ \min(\tcode{(i - first) + 1}, \ \tcode{(last - first) - 1}) \]
applications of the corresponding predicate,
where \tcode{i} is \tcode{adjacent_find}'s return value.
For the overloads with an \tcode{ExecutionPolicy},
\bigoh{\tcode{last - first}} applications of the corresponding predicate,
and no more than twice as many applications of any projection.
\end{itemdescr}

\rSec2[alg.count]{Count}

\indexlibrary{\idxcode{count}}%
\indexlibrary{\idxcode{count_if}}%
\begin{itemdecl}
template<class InputIterator, class T>
  constexpr typename iterator_traits<InputIterator>::difference_type
    count(InputIterator first, InputIterator last, const T& value);
template<class ExecutionPolicy, class ForwardIterator, class T>
  typename iterator_traits<ForwardIterator>::difference_type
    count(ExecutionPolicy&& exec,
          ForwardIterator first, ForwardIterator last, const T& value);

template<class InputIterator, class Predicate>
  constexpr typename iterator_traits<InputIterator>::difference_type
    count_if(InputIterator first, InputIterator last, Predicate pred);
template<class ExecutionPolicy, class ForwardIterator, class Predicate>
  typename iterator_traits<ForwardIterator>::difference_type
    count_if(ExecutionPolicy&& exec,
             ForwardIterator first, ForwardIterator last, Predicate pred);

template<InputIterator I, Sentinel<I> S, class T, class Proj = identity>
  requires IndirectRelation<ranges::equal_to, projected<I, Proj>, const T*>
  constexpr iter_difference_t<I>
    ranges::count(I first, S last, const T& value, Proj proj = {});
template<InputRange R, class T, class Proj = identity>
  requires IndirectRelation<ranges::equal_to, projected<iterator_t<R>, Proj>, const T*>
  constexpr iter_difference_t<iterator_t<R>>
    ranges::count(R&& r, const T& value, Proj proj = {});
template<InputIterator I, Sentinel<I> S, class Proj = identity,
         IndirectUnaryPredicate<projected<I, Proj>> Pred>
  constexpr iter_difference_t<I>
    ranges::count_if(I first, S last, Pred pred, Proj proj = {});
template<InputRange R, class Proj = identity,
         IndirectUnaryPredicate<projected<iterator_t<R>, Proj>> Pred>
  constexpr iter_difference_t<iterator_t<R>>
    ranges::count_if(R&& r, Pred pred, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let $E$ be:
\begin{itemize}
\item
  \tcode{*i == value} for the overloads
  with no parameter \tcode{pred} or \tcode{proj},
\item
  \tcode{pred(*i) != false} for the overloads
  with a parameter \tcode{pred} but no parameter \tcode{proj},
\item
  \tcode{invoke(proj, *i) == value} for the overloads
  with a parameter \tcode{proj} but no parameter \tcode{pred},
\item
  \tcode{invoke(pred, invoke(proj, *i)) != false} for the overloads
  with both parameters \tcode{proj} and \tcode{pred}.
\end{itemize}

\pnum
\effects
Returns the number of iterators \tcode{i} in the range \range{first}{last}
for which $E$ holds.

\pnum
\complexity
Exactly \tcode{last - first} applications
of the corresponding predicate and any projection.
\end{itemdescr}

\rSec2[mismatch]{Mismatch}

\indexlibrary{\idxcode{mismatch}}%
\begin{itemdecl}
template<class InputIterator1, class InputIterator2>
  constexpr pair<InputIterator1, InputIterator2>
    mismatch(InputIterator1 first1, InputIterator1 last1,
             InputIterator2 first2);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2>
  pair<ForwardIterator1, ForwardIterator2>
    mismatch(ExecutionPolicy&& exec,
             ForwardIterator1 first1, ForwardIterator1 last1,
             ForwardIterator2 first2);

template<class InputIterator1, class InputIterator2,
         class BinaryPredicate>
  constexpr pair<InputIterator1, InputIterator2>
    mismatch(InputIterator1 first1, InputIterator1 last1,
             InputIterator2 first2, BinaryPredicate pred);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
         class BinaryPredicate>
  pair<ForwardIterator1, ForwardIterator2>
    mismatch(ExecutionPolicy&& exec,
             ForwardIterator1 first1, ForwardIterator1 last1,
             ForwardIterator2 first2, BinaryPredicate pred);

template<class InputIterator1, class InputIterator2>
  constexpr pair<InputIterator1, InputIterator2>
    mismatch(InputIterator1 first1, InputIterator1 last1,
             InputIterator2 first2, InputIterator2 last2);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2>
  pair<ForwardIterator1, ForwardIterator2>
    mismatch(ExecutionPolicy&& exec,
             ForwardIterator1 first1, ForwardIterator1 last1,
             ForwardIterator2 first2, ForwardIterator2 last2);

template<class InputIterator1, class InputIterator2,
         class BinaryPredicate>
  constexpr pair<InputIterator1, InputIterator2>
    mismatch(InputIterator1 first1, InputIterator1 last1,
             InputIterator2 first2, InputIterator2 last2,
             BinaryPredicate pred);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
         class BinaryPredicate>
  pair<ForwardIterator1, ForwardIterator2>
    mismatch(ExecutionPolicy&& exec,
             ForwardIterator1 first1, ForwardIterator1 last1,
             ForwardIterator2 first2, ForwardIterator2 last2,
             BinaryPredicate pred);

template<InputIterator I1, Sentinel<I1> S1, InputIterator I2, Sentinel<I2> S2,
         class Proj1 = identity, class Proj2 = identity,
         IndirectRelation<projected<I1, Proj1>,
                          projected<I2, Proj2>> Pred = ranges::equal_to>
  constexpr ranges::mismatch_result<I1, I2>
    ranges::mismatch(I1 first1, S1 last1, I2 first2, S2 last2, Pred pred = {},
                     Proj1 proj1 = {}, Proj2 proj2 = {});
template<InputRange R1, InputRange R2,
         class Proj1 = identity, class Proj2 = identity,
         IndirectRelation<projected<iterator_t<R1>, Proj1>,
                          projected<iterator_t<R2>, Proj2>> Pred = ranges::equal_to>
  constexpr ranges::mismatch_result<safe_iterator_t<R1>, safe_iterator_t<R2>>
    ranges::mismatch(R1&& r1, R2&& r2, Pred pred = {},
                     Proj1 proj1 = {}, Proj2 proj2 = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{last2} be \tcode{first2 + (last1 - first1)}
for the overloads with no parameter \tcode{last2} or \tcode{r2}.

\pnum
Let $E$ be:
\begin{itemize}
\setlength{\emergencystretch}{1em}
\item
  \tcode{!(*(first1 + n) == *(first2 + n))}
  for the overloads with no parameter \tcode{pred},
\item
  \tcode{pred(*(first1 + n), *(first2 + n)) == false}
  for the overloads with a parameter \tcode{pred} and
  no parameter \tcode{proj1},
\item
  \tcode{!invoke(pred, invoke(proj1, *(first1 + n)), invoke(proj2, *(first2 + n)))}
  for the overloads with both parameters \tcode{pred} and \tcode{proj1}.
\end{itemize}

\pnum
Let $N$ be $\min(\tcode{last1 - first1}, \ \tcode{last2 - first2})$.

\pnum
\returns
\tcode{\{ first1 + n, first2 + n \}},
where \tcode{n} is the smallest integer in \range{0}{$N$} such that $E$ holds,
or $N$ if no such integer exists.

\pnum
\complexity
At most $N$ applications of the corresponding predicate and any projections.
\end{itemdescr}

\rSec2[alg.equal]{Equal}

\indexlibrary{\idxcode{equal}}%
\begin{itemdecl}
template<class InputIterator1, class InputIterator2>
  constexpr bool equal(InputIterator1 first1, InputIterator1 last1,
                       InputIterator2 first2);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2>
  bool equal(ExecutionPolicy&& exec,
             ForwardIterator1 first1, ForwardIterator1 last1,
             ForwardIterator2 first2);

template<class InputIterator1, class InputIterator2,
         class BinaryPredicate>
  constexpr bool equal(InputIterator1 first1, InputIterator1 last1,
                       InputIterator2 first2, BinaryPredicate pred);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
         class BinaryPredicate>
  bool equal(ExecutionPolicy&& exec,
             ForwardIterator1 first1, ForwardIterator1 last1,
             ForwardIterator2 first2, BinaryPredicate pred);

template<class InputIterator1, class InputIterator2>
  constexpr bool equal(InputIterator1 first1, InputIterator1 last1,
                       InputIterator2 first2, InputIterator2 last2);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2>
  bool equal(ExecutionPolicy&& exec,
             ForwardIterator1 first1, ForwardIterator1 last1,
             ForwardIterator2 first2, ForwardIterator2 last2);

template<class InputIterator1, class InputIterator2,
         class BinaryPredicate>
  constexpr bool equal(InputIterator1 first1, InputIterator1 last1,
                       InputIterator2 first2, InputIterator2 last2,
                       BinaryPredicate pred);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
         class BinaryPredicate>
  bool equal(ExecutionPolicy&& exec,
             ForwardIterator1 first1, ForwardIterator1 last1,
             ForwardIterator2 first2, ForwardIterator2 last2,
             BinaryPredicate pred);

template<InputIterator I1, Sentinel<I1> S1, InputIterator I2, Sentinel<I2> S2,
         class Pred = ranges::equal_to, class Proj1 = identity, class Proj2 = identity>
  requires IndirectlyComparable<I1, I2, Pred, Proj1, Proj2>
  constexpr bool ranges::equal(I1 first1, S1 last1, I2 first2, S2 last2,
                               Pred pred = {},
                               Proj1 proj1 = {}, Proj2 proj2 = {});
template<InputRange R1, InputRange R2, class Pred = ranges::equal_to,
         class Proj1 = identity, class Proj2 = identity>
  requires IndirectlyComparable<iterator_t<R1>, iterator_t<R2>, Pred, Proj1, Proj2>
  constexpr bool ranges::equal(R1&& r1, R2&& r2, Pred pred = {},
                               Proj1 proj1 = {}, Proj2 proj2 = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let:
\begin{itemize}
\item
  \tcode{last2} be \tcode{first2 + (last1 - first1)}
  for the overloads with no parameter \tcode{last2} or \tcode{r2},
\item
  \tcode{pred} be \tcode{equal_to\{\}}
  for the overloads with no parameter \tcode{pred},
\item
  $E$ be:
  \begin{itemize}
  \setlength{\emergencystretch}{1em}
  \item
    \tcode{pred(*i, *(first2 + (i - first1)))}
    for the overloads with no parameter \tcode{proj1},
  \item
    \tcode{invoke(pred, invoke(proj1, *i), invoke(proj2, *(first2 + (i - first1))))}
    for the overloads with parameter \tcode{proj1}.
  \end{itemize}
\end{itemize}

\pnum
\returns
If \tcode{last1 - first1 != last2 - first2}, return \tcode{false}.
Otherwise return \tcode{true}
if $E$ holds for every iterator \tcode{i} in the range \range{first1}{last1}
Otherwise, returns \tcode{false}.

\pnum
\complexity
If the types of \tcode{first1}, \tcode{last1}, \tcode{first2}, and \tcode{last2}:
\begin{itemize}
\item
  meet the
  \oldconcept{RandomAccessIterator} requirements\iref{random.access.iterators}
  for the overloads in namespace \tcode{std}, or
\item
  pairwise model \tcode{SizedSentinel}\iref{iterator.concept.sizedsentinel}
  for the overloads in namespace \tcode{ranges},
\end{itemize}
and \tcode{last1 - first1 != last2 - first2},
then no applications of the corresponding predicate and each projection;
otherwise,
\begin{itemize}
\item
  For the overloads with no \tcode{ExecutionPolicy},
  at most $\min(\tcode{last1 - first1}, \ \tcode{last2 - first2})$
  applications of the corresponding predicate and any projections.
\item
  For the overloads with an \tcode{ExecutionPolicy},
  \bigoh{\min(\tcode{last1 - first1}, \ \tcode{last2 - first2})}
  applications of the corresponding predicate.
\end{itemize}
\end{itemdescr}

\rSec2[alg.is.permutation]{Is permutation}

\indexlibrary{\idxcode{is_permutation}}%
\begin{itemdecl}
template<class ForwardIterator1, class ForwardIterator2>
  constexpr bool is_permutation(ForwardIterator1 first1, ForwardIterator1 last1,
                                ForwardIterator2 first2);
template<class ForwardIterator1, class ForwardIterator2,
         class BinaryPredicate>
  constexpr bool is_permutation(ForwardIterator1 first1, ForwardIterator1 last1,
                                ForwardIterator2 first2, BinaryPredicate pred);
template<class ForwardIterator1, class ForwardIterator2>
  constexpr bool is_permutation(ForwardIterator1 first1, ForwardIterator1 last1,
                                ForwardIterator2 first2, ForwardIterator2 last2);
template<class ForwardIterator1, class ForwardIterator2,
         class BinaryPredicate>
  constexpr bool is_permutation(ForwardIterator1 first1, ForwardIterator1 last1,
                                ForwardIterator2 first2, ForwardIterator2 last2,
                                BinaryPredicate pred);
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
\tcode{ForwardIterator1} and \tcode{ForwardIterator2} shall have
the same value type.
The comparison function shall be an equivalence relation.

\pnum
\remarks
If \tcode{last2} was not given in the argument list,
it denotes \tcode{first2 + (last1 - first1)} below.

\pnum
\returns
If \tcode{last1 - first1 != last2 - first2}, return \tcode{false}.
Otherwise return \tcode{true}
if there exists a permutation of the elements
in the range \range{first2}{first2 + (last1 - first1)},
beginning with \tcode{ForwardIterator2 begin},
such that \tcode{equal(first1, last1, begin)} returns \tcode{true} or
\tcode{equal(first1, last1, begin, pred)} returns \tcode{true};
otherwise, returns \tcode{false}.

\pnum
\complexity
No applications of the corresponding predicate
if \tcode{ForwardIterator1} and \tcode{Forward\-Iter\-ator2}
meet the requirements of random access iterators and
\tcode{last1 - first1 != last2 - first2}.
Otherwise, exactly \tcode{last1 - first1} applications
of the corresponding predicate
if \tcode{equal(\brk{}first1, last1, first2, last2)} would return \tcode{true}
if \tcode{pred} was not given in the argument list or
\tcode{equal(first1, last1, first2, last2, pred)} would return \tcode{true}
if \tcode{pred} was given in the argument list;
otherwise, at worst \bigoh{N^2}, where $N$ has the value \tcode{last1 - first1}.
\end{itemdescr}

\indexlibrary{\idxcode{is_permutation}}%
\begin{itemdecl}
template<ForwardIterator I1, Sentinel<I1> S1, ForwardIterator I2,
         Sentinel<I2> S2, class Pred = ranges::equal_to, class Proj1 = identity,
         class Proj2 = identity>
  requires IndirectlyComparable<I1, I2, Pred, Proj1, Proj2>
  constexpr bool ranges::is_permutation(I1 first1, S1 last1, I2 first2, S2 last2,
                                        Pred pred = {},
                                        Proj1 proj1 = {}, Proj2 proj2 = {});
template<ForwardRange R1, ForwardRange R2, class Pred = ranges::equal_to,
         class Proj1 = identity, class Proj2 = identity>
  requires IndirectlyComparable<iterator_t<R1>, iterator_t<R2>, Pred, Proj1, Proj2>
  constexpr bool ranges::is_permutation(R1&& r1, R2&& r2, Pred pred = {},
                                        Proj1 proj1 = {}, Proj2 proj2 = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
\returns
If \tcode{last1 - first1 != last2 - first2}, return \tcode{false}.
Otherwise return \tcode{true} if there exists a permutation of the elements
in the range \range{first2}{last2}, bounded by \range{pfirst}{plast},
such that
\tcode{ranges::equal(first1, last1, pfirst, plast, pred, proj1, proj2)}
returns \tcode{true};
otherwise, returns \tcode{false}.

\pnum
\complexity
No applications of the corresponding predicate and projections if:
\begin{itemize}
\item \tcode{S1} and \tcode{I1} model \libconcept{SizedSentinel},
\item \tcode{S2} and \tcode{I2} model \libconcept{SizedSentinel}, and
\item \tcode{last1 - first1 != last2 - first2}.
\end{itemize}
Otherwise, exactly \tcode{last1 - first1} applications
of the corresponding predicate and projections
if \tcode{ranges::equal(\brk{}first1, last1, first2, last2, pred, proj1, proj2)}
would return \tcode{true};
otherwise, at worst \bigoh{N^2}, where $N$ has the value \tcode{last1 - first1}.
\end{itemdescr}

\rSec2[alg.search]{Search}

\indexlibrary{\idxcode{search}}%
\begin{itemdecl}
template<class ForwardIterator1, class ForwardIterator2>
  constexpr ForwardIterator1
    search(ForwardIterator1 first1, ForwardIterator1 last1,
           ForwardIterator2 first2, ForwardIterator2 last2);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2>
  ForwardIterator1
    search(ExecutionPolicy&& exec,
           ForwardIterator1 first1, ForwardIterator1 last1,
           ForwardIterator2 first2, ForwardIterator2 last2);

template<class ForwardIterator1, class ForwardIterator2,
         class BinaryPredicate>
  constexpr ForwardIterator1
    search(ForwardIterator1 first1, ForwardIterator1 last1,
           ForwardIterator2 first2, ForwardIterator2 last2,
           BinaryPredicate pred);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
         class BinaryPredicate>
  ForwardIterator1
    search(ExecutionPolicy&& exec,
           ForwardIterator1 first1, ForwardIterator1 last1,
           ForwardIterator2 first2, ForwardIterator2 last2,
           BinaryPredicate pred);
\end{itemdecl}

\begin{itemdescr}
\pnum
\returns
The first iterator \tcode{i} in the range \range{first1}{last1 - (last2-first2)}
such that
for every non-negative integer \tcode{n} less than \tcode{last2 - first2}
the following corresponding conditions hold:
\tcode{*(i + n) == *(first2 + n), pred(*(i + n), *(first2 + n)) != false}.
Returns \tcode{first1} if \range{first2}{last2} is empty,
otherwise returns \tcode{last1} if no such iterator is found.

\pnum
\complexity
At most \tcode{(last1 - first1) * (last2 - first2)} applications
of the corresponding predicate.
\end{itemdescr}

\indexlibrary{\idxcode{search}}%
\begin{itemdecl}
template<ForwardIterator I1, Sentinel<I1> S1, ForwardIterator I2,
         Sentinel<I2> S2, class Pred = ranges::equal_to,
         class Proj1 = identity, class Proj2 = identity>
  requires IndirectlyComparable<I1, I2, Pred, Proj1, Proj2>
  constexpr subrange<I1>
    ranges::search(I1 first1, S1 last1, I2 first2, S2 last2, Pred pred = {},
                   Proj1 proj1 = {}, Proj2 proj2 = {});
template<ForwardRange R1, ForwardRange R2, class Pred = ranges::equal_to,
         class Proj1 = identity, class Proj2 = identity>
  requires IndirectlyComparable<iterator_t<R1>, iterator_t<R2>, Pred, Proj1, Proj2>
  constexpr safe_subrange_t<R1>
    ranges::search(R1&& r1, R2&& r2, Pred pred = {},
                   Proj1 proj1 = {}, Proj2 proj2 = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
\returns
\begin{itemize}
\item
  \tcode{\{i, i + (last2 - first2)\}},
  where \tcode{i} is
  the first iterator in the range \range{first1}{last1 - (last2 - first2)}
  such that
  for every non-negative integer \tcode{n} less than \tcode{last2 - first2}
  the condition
\begin{codeblock}
bool(invoke(pred, invoke(proj1, *(i + n)), invoke(proj2, *(first2 + n))))
\end{codeblock}
  is \tcode{true}:
\item
  Returns \tcode{\{last1, last1\}} if no such iterator exists.
\end{itemize}

\pnum
\complexity
At most \tcode{(last1 - first1) * (last2 - first2)} applications
of the corresponding predicate and projections.
\end{itemdescr}

\indexlibrary{\idxcode{search_n}}%
\begin{itemdecl}
template<class ForwardIterator, class Size, class T>
  constexpr ForwardIterator
    search_n(ForwardIterator first, ForwardIterator last,
             Size count, const T& value);
template<class ExecutionPolicy, class ForwardIterator, class Size, class T>
  ForwardIterator
    search_n(ExecutionPolicy&& exec,
             ForwardIterator first, ForwardIterator last,
             Size count, const T& value);

template<class ForwardIterator, class Size, class T,
         class BinaryPredicate>
  constexpr ForwardIterator
    search_n(ForwardIterator first, ForwardIterator last,
             Size count, const T& value,
             BinaryPredicate pred);
template<class ExecutionPolicy, class ForwardIterator, class Size, class T,
         class BinaryPredicate>
  ForwardIterator
    search_n(ExecutionPolicy&& exec,
             ForwardIterator first, ForwardIterator last,
             Size count, const T& value,
             BinaryPredicate pred);
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
The type \tcode{Size}
shall be convertible to integral type~(\ref{conv.integral}, \ref{class.conv}).

\pnum
\returns
The first iterator \tcode{i} in the range \range{first}{last-count}
such that for every non-negative integer \tcode{n} less than \tcode{count}
the following corresponding conditions hold:
\tcode{*(i + n) == value, pred(*(i + n),value) != false}.
Returns \tcode{last} if no such iterator is found.

\pnum
\complexity
At most \tcode{last - first} applications of the corresponding predicate.
\end{itemdescr}

\indexlibrary{\idxcode{search_n}}%
\begin{itemdecl}
template<ForwardIterator I, Sentinel<I> S, class T,
         class Pred = ranges::equal_to, class Proj = identity>
  requires IndirectlyComparable<I, const T*, Pred, Proj>
  constexpr subrange<I>
    ranges::search_n(I first, S last, iter_difference_t<I> count,
                     const T& value, Pred pred = {}, Proj proj = {});
template<ForwardRange R, class T, class Pred = ranges::equal_to,
         class Proj = identity>
  requires IndirectlyComparable<iterator_t<R>, const T*, Pred, Proj>
  constexpr safe_subrange_t<R>
    ranges::search_n(R&& r, iter_difference_t<iterator_t<R>> count,
                     const T& value, Pred pred = {}, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
\returns
\tcode{\{i, i + count\}}
where \tcode{i} is the first iterator in the range \range{first}{last - count}
such that for every non-negative integer \tcode{n} less than \tcode{count},
the following condition holds:
\tcode{invoke(pred, invoke(proj, *(i + n)), value)}.
Returns \tcode{\{last, last\}} if no such iterator is found.

\pnum
\complexity
At most \tcode{last - first} applications
of the corresponding predicate and projection.
\end{itemdescr}

\indexlibrary{\idxcode{search}}%
\begin{itemdecl}
template<class ForwardIterator, class Searcher>
  constexpr ForwardIterator
    search(ForwardIterator first, ForwardIterator last, const Searcher& searcher);
\end{itemdecl}

\begin{itemdescr}
\pnum
\effects
Equivalent to: \tcode{return searcher(first, last).first;}

\pnum
\remarks
\tcode{Searcher} need not meet the \oldconcept{CopyConstructible} requirements.
\end{itemdescr}

\rSec1[alg.modifying.operations]{Mutating sequence operations}

\rSec2[alg.copy]{Copy}

\indexlibrary{\idxcode{copy}}%
\begin{itemdecl}
template<class InputIterator, class OutputIterator>
  constexpr OutputIterator copy(InputIterator first, InputIterator last,
                                OutputIterator result);

template<InputIterator I, Sentinel<I> S, WeaklyIncrementable O>
  requires IndirectlyCopyable<I, O>
  constexpr ranges::copy_result<I, O> ranges::copy(I first, S last, O result);
template<InputRange R, WeaklyIncrementable O>
  requires IndirectlyCopyable<iterator_t<R>, O>
  constexpr ranges::copy_result<safe_iterator_t<R>, O> ranges::copy(R&& r, O result);
\end{itemdecl}

\begin{itemdescr}
\pnum
Let $N$ be \tcode{last - first}.

\pnum
\requires
\tcode{result} shall not be in the range \range{first}{last}.

\pnum
\effects
Copies elements in the range \range{first}{last}
into the range \range{result}{result + $N$}
starting from \tcode{first} and proceeding to \tcode{last}.
For each non-negative integer $n < N$,
performs \tcode{*(result + $n$) = *(first + $n$)}.

\pnum
\returns
\begin{itemize}
\item
  \tcode{result + $N$} for the overload in namespace \tcode{std}, or
\item
  \tcode{\{last, result + $N$\}} for the overloads in namespace \tcode{ranges}.
\end{itemize}

\pnum
\complexity
Exactly $N$ assignments.
\end{itemdescr}

\indexlibrary{\idxcode{copy}}%
\begin{itemdecl}
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2>
  ForwardIterator2 copy(ExecutionPolicy&& policy,
                        ForwardIterator1 first, ForwardIterator1 last,
                        ForwardIterator2 result);
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
The ranges \range{first}{last} and \range{result}{result + (last - first)}
shall not overlap.

\pnum
\effects
Copies elements in the range \range{first}{last}
into the range \range{result}{result + (last - first)}.
For each non-negative integer \tcode{n < (last - first)},
performs \tcode{*(result + n) = *(first + n)}.

\pnum
\returns
\tcode{result + (last - first)}.

\pnum
\complexity
Exactly \tcode{last - first} assignments.
\end{itemdescr}

\indexlibrary{\idxcode{copy_n}}%
\begin{itemdecl}
template<class InputIterator, class Size, class OutputIterator>
  constexpr OutputIterator copy_n(InputIterator first, Size n,
                                  OutputIterator result);
template<class ExecutionPolicy, class ForwardIterator1, class Size, class ForwardIterator2>
  ForwardIterator2 copy_n(ExecutionPolicy&& exec,
                          ForwardIterator1 first, Size n,
                          ForwardIterator2 result);

template<InputIterator I, WeaklyIncrementable O>
  requires IndirectlyCopyable<I, O>
  constexpr ranges::copy_n_result<I, O>
    ranges::copy_n(I first, iter_difference_t<I> n, O result);
\end{itemdecl}

\begin{itemdescr}
\pnum
Let $M$ be $\max(\tcode{n}, 0)$.

\pnum
\effects
For each non-negative integer $i < M$,
performs \tcode{*(result + i) = *(first + i)}.

\pnum
\returns
\begin{itemize}
\item
  \tcode{result + $M$}
  for the overloads in namespace \tcode{std}, or
\item
  \tcode{\{first + $M$, result + $M$\}}
  for the overload in namespace \tcode{ranges}.
\end{itemize}

\pnum
\complexity
Exactly $M$ assignments.
\end{itemdescr}

\indexlibrary{\idxcode{copy_if}}%
\begin{itemdecl}
template<class InputIterator, class OutputIterator, class Predicate>
  constexpr OutputIterator copy_if(InputIterator first, InputIterator last,
                                   OutputIterator result, Predicate pred);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
         class Predicate>
  ForwardIterator2 copy_if(ExecutionPolicy&& exec,
                           ForwardIterator1 first, ForwardIterator1 last,
                           ForwardIterator2 result, Predicate pred);

template<InputIterator I, Sentinel<I> S, WeaklyIncrementable O, class Proj = identity,
         IndirectUnaryPredicate<projected<I, Proj>> Pred>
  requires IndirectlyCopyable<I, O>
  constexpr ranges::copy_if_result<I, O>
    ranges::copy_if(I first, S last, O result, Pred pred, Proj proj = {});
template<InputRange R, WeaklyIncrementable O, class Proj = identity,
         IndirectUnaryPredicate<projected<iterator_t<R>, Proj>> Pred>
  requires IndirectlyCopyable<iterator_t<R>, O>
  constexpr ranges::copy_if_result<safe_iterator_t<R>, O>
    ranges::copy_if(R&& r, O result, Pred pred, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let $E$ be:
\begin{itemize}
\item
  \tcode{bool(pred(*i))}
  for the overloads in namespace \tcode{std}, or
\item
  \tcode{bool(invoke(pred, invoke(proj, *i)))}
  for the overloads in namespace \tcode{ranges}.
\end{itemize}
and $N$ be the number of iterators \tcode{i} in the range \range{first}{last}
for which the condition $E$ holds.

\pnum
\requires
The ranges \range{first}{last} and \range{result}{result + (last - first)}
shall not overlap.
\begin{note}
For the overload with an \tcode{ExecutionPolicy},
there may be a performance cost
if \tcode{iterator_traits<ForwardIterator1>::value_type}
is not \oldconcept{\-Move\-Constructible} (\tref{cpp17.moveconstructible}).
\end{note}

\pnum
\effects
Copies all of the elements referred to
by the iterator \tcode{i} in the range \range{first}{last}
for which $E$ is \tcode{true}.

\pnum
\returns
\begin{itemize}
\item
  \tcode{result + $N$}
  for the overloads in namespace \tcode{std}, or
\item
  \tcode{\{last, result + $N$\}}
  for the overloads in namespace \tcode{ranges}.
\end{itemize}

\pnum
\complexity
Exactly \tcode{last - first} applications
of the corresponding predicate and any projection.

\pnum
\remarks
Stable\iref{algorithm.stable}.
\end{itemdescr}

\indexlibrary{\idxcode{copy_backward}}%
\begin{itemdecl}
template<class BidirectionalIterator1, class BidirectionalIterator2>
  constexpr BidirectionalIterator2
    copy_backward(BidirectionalIterator1 first,
                  BidirectionalIterator1 last,
                  BidirectionalIterator2 result);

template<BidirectionalIterator I1, Sentinel<I1> S1, BidirectionalIterator I2>
  requires IndirectlyCopyable<I1, I2>
  constexpr ranges::copy_backward_result<I1, I2>
    ranges::copy_backward(I1 first, S1 last, I2 result);
template<BidirectionalRange R, BidirectionalIterator I>
  requires IndirectlyCopyable<iterator_t<R>, I>
  constexpr ranges::copy_backward_result<safe_iterator_t<R>, I>
    ranges::copy_backward(R&& r, I result);
\end{itemdecl}

\begin{itemdescr}
\pnum
Let $N$ be \tcode{last - first}.

\pnum
\requires
\tcode{result} shall not be in the range \brange{first}{last}.

\pnum
\effects
Copies elements in the range \range{first}{last}
into the range \range{result - $N$}{result}
starting from \tcode{last - 1} and proceeding to \tcode{first}.%
\footnote{\tcode{copy_backward} should be used instead of copy
when \tcode{last} is in the range \range{result - $N$}{result}.}
For each positive integer $n \le N$,
performs \tcode{*(result - $n$) = *(last - $n$)}.

\pnum
\returns
\begin{itemize}
\item
  \tcode{result - $N$}
  for the overload in namespace \tcode{std}, or
\item
  \tcode{\{last, result - $N$\}}
  for the overloads in namespace \tcode{ranges}.
\end{itemize}

\pnum
\complexity
Exactly $N$ assignments.
\end{itemdescr}

\rSec2[alg.move]{Move}

\indexlibrary{\idxcode{move}!algorithm}%
\begin{itemdecl}
template<class InputIterator, class OutputIterator>
  constexpr OutputIterator move(InputIterator first, InputIterator last,
                                OutputIterator result);

template<InputIterator I, Sentinel<I> S, WeaklyIncrementable O>
  requires IndirectlyMovable<I, O>
  constexpr ranges::move_result<I, O>
    ranges::move(I first, S last, O result);
template<InputRange R, WeaklyIncrementable O>
  requires IndirectlyMovable<iterator_t<R>, O>
  constexpr ranges::move_result<safe_iterator_t<R>, O>
    ranges::move(R&& r, O result);
\end{itemdecl}

\begin{itemdescr}
\pnum
Let $E$ be
\begin{itemize}
\item
  \tcode{std::move(*(first + $n$))}
  for the overload in namespace \tcode{std}, or
\item
  \tcode{ranges::iter_move(first + $n$)}
  for the overloads in namespace \tcode{ranges}.
\end{itemize}
Let $N$ be \tcode{last - first}.

\pnum
\requires
\tcode{result} shall not be in the range \range{first}{last}.

\pnum
\effects
Moves elements in the range \range{first}{last}
into the range \range{result}{result + $N$}
starting from \tcode{first} and proceeding to \tcode{last}.
For each non-negative integer $n < N$, performs \tcode{*(result + $n$) = $E$}.

\pnum
\returns
\begin{itemize}
\item
  \tcode{result + $N$}
  for the overload in namespace \tcode{std}, or
\item
  \tcode{\{last, result + $N$\}}
  for the overloads in namespace \tcode{ranges}.
\end{itemize}

\pnum
\complexity
Exactly $N$ assignments.
\end{itemdescr}

\indexlibrary{\idxcode{move}!algorithm}%
\begin{itemdecl}
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2>
  ForwardIterator2 move(ExecutionPolicy&& policy,
                        ForwardIterator1 first, ForwardIterator1 last,
                        ForwardIterator2 result);
\end{itemdecl}

\begin{itemdescr}
\pnum
Let $N$ be \tcode{last - first}.

\pnum
\requires
The ranges \range{first}{last} and \range{result}{result + $N$}
shall not overlap.

\pnum
\effects
Moves elements in the range \range{first}{last}
into the range \range{result}{result + $N$}.
For each non-negative integer $n < N$,
performs \tcode{*(result + $n$) = std::\brk{}move(*(first + $n$))}.

\pnum
\returns
\tcode{result + $N$}.

\pnum
\complexity
Exactly $N$ assignments.
\end{itemdescr}

\indexlibrary{\idxcode{move_backward}}%
\begin{itemdecl}
template<class BidirectionalIterator1, class BidirectionalIterator2>
  constexpr BidirectionalIterator2
    move_backward(BidirectionalIterator1 first, BidirectionalIterator1 last,
                  BidirectionalIterator2 result);

template<BidirectionalIterator I1, Sentinel<I1> S1, BidirectionalIterator I2>
  requires IndirectlyMovable<I1, I2>
  constexpr ranges::move_backward_result<I1, I2>
    ranges::move_backward(I1 first, S1 last, I2 result);
template<BidirectionalRange R, BidirectionalIterator I>
  requires IndirectlyMovable<iterator_t<R>, I>
  constexpr ranges::move_backward_result<safe_iterator_t<R>, I>
    ranges::move_backward(R&& r, I result);
\end{itemdecl}

\begin{itemdescr}
\pnum
Let $E$ be
\begin{itemize}
\item
  \tcode{std::move(*(last - $n$))}
  for the overload in namespace \tcode{std}, or
\item
  \tcode{ranges::iter_move(last - $n$)}
  for the overloads in namespace \tcode{ranges}.
\end{itemize}
Let $N$ be \tcode{last - first}.

\pnum
\requires
\tcode{result} shall not be in the range \brange{first}{last}.

\pnum
\effects
Moves elements in the range \range{first}{last}
into the range \range{result - $N$}{result}
starting from \tcode{last - 1} and proceeding to \tcode{first}.%
\footnote{\tcode{move_backward} should be used instead of move
when \tcode{last} is in the range \range{result - $N$}{result}.}
For each positive integer $n \le N$,
performs \tcode{*(result - $n$) = $E$}.

\pnum
\returns
\begin{itemize}
\item
  \tcode{result - $N$}
  for the overload in namespace \tcode{std}, or
\item
  \tcode{\{last, result - $N$\}}
  for the overloads in namespace \tcode{ranges}.
\end{itemize}

\pnum
\complexity
Exactly $N$ assignments.
\end{itemdescr}

\rSec2[alg.swap]{Swap}

\indexlibrary{\idxcode{swap_ranges}}%
\begin{itemdecl}
template<class ForwardIterator1, class ForwardIterator2>
  constexpr ForwardIterator2
    swap_ranges(ForwardIterator1 first1, ForwardIterator1 last1,
                ForwardIterator2 first2);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2>
  ForwardIterator2
    swap_ranges(ExecutionPolicy&& exec,
                ForwardIterator1 first1, ForwardIterator1 last1,
                ForwardIterator2 first2);

template<InputIterator I1, Sentinel<I1> S1, InputIterator I2, Sentinel<I2> S2>
  requires IndirectlySwappable<I1, I2>
  constexpr ranges::swap_ranges_result<I1, I2>
    ranges::swap_ranges(I1 first1, S1 last1, I2 first2, S2 last2);
template<InputRange R1, InputRange R2>
  requires IndirectlySwappable<iterator_t<R1>, iterator_t<R2>>
  constexpr ranges::swap_ranges_result<safe_iterator_t<R1>, safe_iterator_t<R2>>
    ranges::swap_ranges(R1&& r1, R2&& r2);
\end{itemdecl}

\begin{itemdescr}
\pnum
Let:
\begin{itemize}
\item
  \tcode{last2} be \tcode{first2 + (last1 - first1)}
  for the overloads with no parameter named \tcode{last2}, and
\item $M$ be $\min(\tcode{last1 - first1}, \ \tcode{last2 - first2})$.
\end{itemize}

\pnum
\requires
The two ranges \range{first1}{last1} and \range{first2}{last2}
shall not overlap.
For the overloads in namespace \tcode{std},
\tcode{*(first1 + $n$)} shall be swappable with\iref{swappable.requirements}
\tcode{*(first2 + $n$)}.

\pnum
\effects
For each non-negative integer $n < M$ performs:
\begin{itemize}
\item
  \tcode{swap(*(first1 + $n$), *(first2 + $n$))}
  for the overloads in namespace \tcode{std}, or
\item
  \tcode{ranges::iter_swap(first1 + $n$, first2 + $n$)}
  for the overloads in namespace \tcode{ranges}.
\end{itemize}

\pnum
\returns
\begin{itemize}
\item
  \tcode{last2}
  for the overloads in namespace \tcode{std}, or
\item
  \tcode{\{first1 + $M$, first2 + $M$\}}
  for the overloads in namespace \tcode{ranges}.
\end{itemize}

\pnum
\complexity
Exactly $M$ swaps.
\end{itemdescr}

\indexlibrary{\idxcode{iter_swap}}%
\begin{itemdecl}
template<class ForwardIterator1, class ForwardIterator2>
  constexpr void iter_swap(ForwardIterator1 a, ForwardIterator2 b);
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
\tcode{a} and \tcode{b} shall be dereferenceable. \tcode{*a} shall be
swappable with\iref{swappable.requirements} \tcode{*b}.

\pnum
\effects
As if by \tcode{swap(*a, *b)}.
\end{itemdescr}

\rSec2[alg.transform]{Transform}

\indexlibrary{\idxcode{transform}}%
\begin{itemdecl}
template<class InputIterator, class OutputIterator,
         class UnaryOperation>
  constexpr OutputIterator
    transform(InputIterator first1, InputIterator last1,
              OutputIterator result, UnaryOperation op);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
         class UnaryOperation>
  ForwardIterator2
    transform(ExecutionPolicy&& exec,
              ForwardIterator1 first1, ForwardIterator1 last1,
              ForwardIterator2 result, UnaryOperation op);

template<class InputIterator1, class InputIterator2,
         class OutputIterator, class BinaryOperation>
  constexpr OutputIterator
    transform(InputIterator1 first1, InputIterator1 last1,
              InputIterator2 first2, OutputIterator result,
              BinaryOperation binary_op);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
         class ForwardIterator, class BinaryOperation>
  ForwardIterator
    transform(ExecutionPolicy&& exec,
              ForwardIterator1 first1, ForwardIterator1 last1,
              ForwardIterator2 first2, ForwardIterator result,
              BinaryOperation binary_op);

template<InputIterator I, Sentinel<I> S, WeaklyIncrementable O,
         CopyConstructible F, class Proj = identity>
  requires Writable<O, indirect_result_t<F&, projected<I, Proj>>>
  constexpr ranges::unary_transform_result<I, O>
    ranges::transform(I first1, S last1, O result, F op, Proj proj = {});
template<InputRange R, WeaklyIncrementable O, CopyConstructible F,
         class Proj = identity>
  requires Writable<O, indirect_result_t<F&, projected<iterator_t<R>, Proj>>>
  constexpr ranges::unary_transform_result<safe_iterator_t<R>, O>
    ranges::transform(R&& r, O result, F op, Proj proj = {});
template<InputIterator I1, Sentinel<I1> S1, InputIterator I2, Sentinel<I2> S2,
         WeaklyIncrementable O, CopyConstructible F, class Proj1 = identity,
         class Proj2 = identity>
  requires Writable<O, indirect_result_t<F&, projected<I1, Proj1>,
                                         projected<I2, Proj2>>>
  constexpr ranges::binary_transform_result<I1, I2, O>
    ranges::transform(I1 first1, S1 last1, I2 first2, S2 last2, O result,
                      F binary_op, Proj1 proj1 = {}, Proj2 proj2 = {});
template<InputRange R1, InputRange R2, WeaklyIncrementable O,
         CopyConstructible F, class Proj1 = identity, class Proj2 = identity>
  requires Writable<O, indirect_result_t<F&, projected<iterator_t<R1>, Proj1>,
                                         projected<iterator_t<R2>, Proj2>>>
  constexpr ranges::binary_transform_result<safe_iterator_t<R1>, safe_iterator_t<R2>, O>
    ranges::transform(R1&& r1, R2&& r2, O result,
                      F binary_op, Proj1 proj1 = {}, Proj2 proj2 = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let:
\begin{itemize}
\setlength{\emergencystretch}{1em}
\item
  \tcode{last2} be \tcode{first2 + (last1 - first1)}
  for the overloads with parameter \tcode{first2}
  but no parameter \tcode{last2},
\item
  $N$ be \tcode{last1 - first1} for unary transforms, or
  $\min(\tcode{last1 - first1}, \ \tcode{last2 - first2})$ for binary transforms, and
\item
  $E$ be
  \begin{itemize}
  \item
    \tcode{op(*(first1 + (i - result)))}
    for unary transforms defined in namespace \tcode{std},
  \item
    \tcode{binary_op(*(first1 + (i - result)), *(first2 + (i - result)))}
    for binary transforms defined in namespace \tcode{std},
  \item
    \tcode{invoke(op, invoke(proj, *(first1 + (i - result))))}
    for unary transforms defined in namespace \tcode{ranges}, or
  \item
    \tcode{invoke(binary_op, invoke(proj1, *(first1 + (i - result))), invoke(proj2,\linebreak *(first2 + (i - result))))}
    for binary transforms defined in namespace \tcode{ranges}.
  \end{itemize}
\end{itemize}

\pnum
\requires
\tcode{op} and \tcode{binary_op} shall not invalidate iterators or subranges, nor
modify elements in the ranges
\begin{itemize}
\item \crange{first1}{first1 + $N$},
\item \crange{first2}{first2 + $N$}, and
\item \crange{result}{result + $N$}.%
\footnote{The use of fully closed ranges is intentional.}
\end{itemize}

\pnum
\effects
Assigns through every iterator \tcode{i}
in the range \range{result}{result + $N$}
a new corresponding value equal to $E$.

\pnum
\returns
\begin{itemize}
\item
  \tcode{result + $N$}
  for the overloads defined in namespace \tcode{std},
\item
  \tcode{\{first1 + $N$, result + $N$\}}
  for unary transforms defined in namespace \tcode{ranges}, or
\item \tcode{\{first1 + $N$, first2 + $N$, result + $N$\}}
  for binary transforms defined in namespace \tcode{ranges}.
\end{itemize}

\pnum
\complexity
Exactly $N$ applications of \tcode{op} or \tcode{binary_op}, and
any projections.
This requirement also applies to the overload with an \tcode{ExecutionPolicy}.

\pnum
\remarks
\tcode{result} may be equal to \tcode{first1} or \tcode{first2}.
\end{itemdescr}

\rSec2[alg.replace]{Replace}

\indexlibrary{\idxcode{replace}}%
\indexlibrary{\idxcode{replace_if}}%
\begin{itemdecl}
template<class ForwardIterator, class T>
  constexpr void replace(ForwardIterator first, ForwardIterator last,
                         const T& old_value, const T& new_value);
template<class ExecutionPolicy, class ForwardIterator, class T>
  void replace(ExecutionPolicy&& exec,
               ForwardIterator first, ForwardIterator last,
               const T& old_value, const T& new_value);

template<class ForwardIterator, class Predicate, class T>
  constexpr void replace_if(ForwardIterator first, ForwardIterator last,
                            Predicate pred, const T& new_value);
template<class ExecutionPolicy, class ForwardIterator, class Predicate, class T>
  void replace_if(ExecutionPolicy&& exec,
                  ForwardIterator first, ForwardIterator last,
                  Predicate pred, const T& new_value);

template<InputIterator I, Sentinel<I> S, class T1, class T2, class Proj = identity>
  requires Writable<I, const T2&> &&
           IndirectRelation<ranges::equal_to, projected<I, Proj>, const T1*>
  constexpr I
    ranges::replace(I first, S last, const T1& old_value, const T2& new_value, Proj proj = {});
template<InputRange R, class T1, class T2, class Proj = identity>
  requires Writable<iterator_t<R>, const T2&> &&
           IndirectRelation<ranges::equal_to, projected<iterator_t<R>, Proj>, const T1*>
  constexpr safe_iterator_t<R>
    ranges::replace(R&& r, const T1& old_value, const T2& new_value, Proj proj = {});
template<InputIterator I, Sentinel<I> S, class T, class Proj = identity,
         IndirectUnaryPredicate<projected<I, Proj>> Pred>
  requires Writable<I, const T&>
  constexpr I ranges::replace_if(I first, S last, Pred pred, const T& new_value, Proj proj = {});
template<InputRange R, class T, class Proj = identity,
         IndirectUnaryPredicate<projected<iterator_t<R>, Proj>> Pred>
  requires Writable<iterator_t<R>, const T&>
  constexpr safe_iterator_t<R>
    ranges::replace_if(R&& r, Pred pred, const T& new_value, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let $E$ be
\begin{itemize}
\item \tcode{bool(*i == old_value)} for \tcode{replace},
\item \tcode{bool(pred(*i))} for \tcode{replace_if},
\item \tcode{bool(invoke(proj, *i) == old_value)} for \tcode{ranges::replace}, or
\item \tcode{bool(invoke(pred, invoke(proj, *i)))} for \tcode{ranges::replace_if}.
\end{itemize}

\pnum
\requires
The expression \tcode{*first = new_value} shall be valid.

\pnum
\effects
Substitutes elements referred by the iterator \tcode{i}
in the range \range{first}{last} with \tcode{new_value},
when $E$ is \tcode{true}.

\pnum
\returns
\tcode{last} for the overloads in namespace \tcode{ranges}.

\pnum
\complexity
Exactly \tcode{last - first} applications
of the corresponding predicate and any projection.
\end{itemdescr}

\indexlibrary{\idxcode{replace_copy}}%
\indexlibrary{\idxcode{replace_copy_if}}%
\begin{itemdecl}
template<class InputIterator, class OutputIterator, class T>
  constexpr OutputIterator
    replace_copy(InputIterator first, InputIterator last,
                 OutputIterator result,
                 const T& old_value, const T& new_value);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2, class T>
  ForwardIterator2
    replace_copy(ExecutionPolicy&& exec,
                 ForwardIterator1 first, ForwardIterator1 last,
                 ForwardIterator2 result,
                 const T& old_value, const T& new_value);

template<class InputIterator, class OutputIterator, class Predicate, class T>
  constexpr OutputIterator
    replace_copy_if(InputIterator first, InputIterator last,
                    OutputIterator result,
                    Predicate pred, const T& new_value);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
         class Predicate, class T>
  ForwardIterator2
    replace_copy_if(ExecutionPolicy&& exec,
                    ForwardIterator1 first, ForwardIterator1 last,
                    ForwardIterator2 result,
                    Predicate pred, const T& new_value);

template<InputIterator I, Sentinel<I> S, class T1, class T2, OutputIterator<const T2&> O,
         class Proj = identity>
  requires IndirectlyCopyable<I, O> &&
           IndirectRelation<ranges::equal_to, projected<I, Proj>, const T1*>
  constexpr ranges::replace_copy_result<I, O>
    ranges::replace_copy(I first, S last, O result, const T1& old_value, const T2& new_value,
                         Proj proj = {});
template<InputRange R, class T1, class T2, OutputIterator<const T2&> O,
         class Proj = identity>
  requires IndirectlyCopyable<iterator_t<R>, O> &&
           IndirectRelation<ranges::equal_to, projected<iterator_t<R>, Proj>, const T1*>
  constexpr ranges::replace_copy_result<safe_iterator_t<R>, O>
    ranges::replace_copy(R&& r, O result, const T1& old_value, const T2& new_value,
                         Proj proj = {});

template<InputIterator I, Sentinel<I> S, class T, OutputIterator<const T&> O,
         class Proj = identity, IndirectUnaryPredicate<projected<I, Proj>> Pred>
  requires IndirectlyCopyable<I, O>
  constexpr ranges::replace_copy_if_result<I, O>
    ranges::replace_copy_if(I first, S last, O result, Pred pred, const T& new_value,
                            Proj proj = {});
template<InputRange R, class T, OutputIterator<const T&> O, class Proj = identity,
         IndirectUnaryPredicate<projected<iterator_t<R>, Proj>> Pred>
  requires IndirectlyCopyable<iterator_t<R>, O>
  constexpr ranges::replace_copy_if_result<safe_iterator_t<R>, O>
    ranges::replace_copy_if(R&& r, O result, Pred pred, const T& new_value,
                            Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\setlength{\emergencystretch}{1.5em}
\pnum
Let $E$ be
\begin{itemize}
\item \tcode{bool(*(first + (i - result)) == old_value)}
  for \tcode{replace_copy},
\item \tcode{bool(pred(*(first + (i - result))))}
  for \tcode{replace_copy_if},
\item \tcode{bool(invoke(proj, *(first + (i - result))) == old_value)}
  for \tcode{ranges::replace_copy},
\item \tcode{bool(invoke(pred, invoke(proj, *(first + (i - result)))))}
  for \tcode{ranges::replace_\-copy_if}.
\end{itemize}

\pnum
\requires
The results of the expressions \tcode{*first} and \tcode{new_value}
shall be writable\iref{iterator.requirements.general}
to the \tcode{result} output iterator.
The ranges \range{first}{last} and \range{result}{result + (last - first)}
shall not overlap.

\pnum
\effects
Assigns to every iterator \tcode{i}
in the range \range{result}{result + (last - first)}
either \tcode{new_value} or \tcode{*\brk(first + (i - result))}
depending on whether $E$ holds.

\pnum
\returns
\begin{itemize}
\item
  \tcode{result + (last - first)}
  for the overloads in namespace \tcode{std}, or
\item
  \tcode{\{last, result + (last - first)\}}
  for the overloads in namespace \tcode{ranges}.
\end{itemize}

\pnum
\complexity
Exactly \tcode{last - first} applications
of the corresponding predicate and any projection.
\end{itemdescr}

\rSec2[alg.fill]{Fill}

\indexlibrary{\idxcode{fill}}%
\indexlibrary{\idxcode{fill_n}}%
\begin{itemdecl}
template<class ForwardIterator, class T>
  constexpr void fill(ForwardIterator first, ForwardIterator last, const T& value);
template<class ExecutionPolicy, class ForwardIterator, class T>
  void fill(ExecutionPolicy&& exec,
            ForwardIterator first, ForwardIterator last, const T& value);

template<class OutputIterator, class Size, class T>
  constexpr OutputIterator fill_n(OutputIterator first, Size n, const T& value);
template<class ExecutionPolicy, class ForwardIterator, class Size, class T>
  ForwardIterator fill_n(ExecutionPolicy&& exec,
                         ForwardIterator first, Size n, const T& value);


template<class T, OutputIterator<const T&> O, Sentinel<O> S>
  constexpr O ranges::fill(O first, S last, const T& value);
template<class T, OutputRange<const T&> R>
  constexpr safe_iterator_t<R> ranges::fill(R&& r, const T& value);
template<class T, OutputIterator<const T&> O>
  constexpr O ranges::fill_n(O first, iter_difference_t<O> n, const T& value);
\end{itemdecl}

\begin{itemdescr}
\pnum
Let $N$ be $\max(0, \tcode{n})$ for the \tcode{fill_n} algorithms, and
\tcode{last - first} for the \tcode{fill} algorithms.

\pnum
\requires
The expression \tcode{value}
shall be writable\iref{iterator.requirements.general} to the output iterator.
The type \tcode{Size} shall be convertible
to an integral type~(\ref{conv.integral}, \ref{class.conv}).

\pnum
\effects
Assigns \tcode{value}
through all the iterators in the range \range{first}{first + $N$}.

\pnum
\returns
\tcode{first + $N$}.

\pnum
\complexity
Exactly $N$ assignments.
\end{itemdescr}

\rSec2[alg.generate]{Generate}

\indexlibrary{\idxcode{generate}}%
\indexlibrary{\idxcode{generate_n}}%
\begin{itemdecl}
template<class ForwardIterator, class Generator>
  constexpr void generate(ForwardIterator first, ForwardIterator last,
                          Generator gen);
template<class ExecutionPolicy, class ForwardIterator, class Generator>
  void generate(ExecutionPolicy&& exec,
                ForwardIterator first, ForwardIterator last,
                Generator gen);

template<class OutputIterator, class Size, class Generator>
  constexpr OutputIterator generate_n(OutputIterator first, Size n, Generator gen);
template<class ExecutionPolicy, class ForwardIterator, class Size, class Generator>
  ForwardIterator generate_n(ExecutionPolicy&& exec,
                             ForwardIterator first, Size n, Generator gen);

template<Iterator O, Sentinel<O> S, CopyConstructible F>
  requires Invocable<F&> && Writable<O, invoke_result_t<F&>>
  constexpr O ranges::generate(O first, S last, F gen);
template<class R, CopyConstructible F>
  requires Invocable<F&> && OutputRange<R, invoke_result_t<F&>>
  constexpr safe_iterator_t<R> ranges::generate(R&& r, F gen);
template<Iterator O, CopyConstructible F>
  requires Invocable<F&> && Writable<O, invoke_result_t<F&>>
  constexpr O ranges::generate_n(O first, iter_difference_t<O> n, F gen);
\end{itemdecl}

\begin{itemdescr}
\pnum
Let $N$ be $\max(0, \tcode{n})$ for the \tcode{generate_n} algorithms, and
\tcode{last - first} for the \tcode{generate} algorithms.

\pnum
\requires
\tcode{Size} shall be convertible
to an integral type~(\ref{conv.integral}, \ref{class.conv}).

\pnum
\effects
Assigns the result of successive evaluations of \tcode{gen()}
through each iterator in the range \range{first}{first + $N$}.

\pnum
\returns
\tcode{first + $N$}.

\pnum
\complexity
Exactly $N$ evaluations of \tcode{gen()} and assignments.
\end{itemdescr}

\rSec2[alg.remove]{Remove}

\indexlibrary{\idxcode{remove}}%
\indexlibrary{\idxcode{remove_if}}%
\begin{itemdecl}
template<class ForwardIterator, class T>
  constexpr ForwardIterator remove(ForwardIterator first, ForwardIterator last,
                                   const T& value);
template<class ExecutionPolicy, class ForwardIterator, class T>
  ForwardIterator remove(ExecutionPolicy&& exec,
                         ForwardIterator first, ForwardIterator last,
                         const T& value);

template<class ForwardIterator, class Predicate>
  constexpr ForwardIterator remove_if(ForwardIterator first, ForwardIterator last,
                                      Predicate pred);
template<class ExecutionPolicy, class ForwardIterator, class Predicate>
  ForwardIterator remove_if(ExecutionPolicy&& exec,
                            ForwardIterator first, ForwardIterator last,
                            Predicate pred);

template<Permutable I, Sentinel<I> S, class T, class Proj = identity>
  requires IndirectRelation<ranges::equal_to, projected<I, Proj>, const T*>
  constexpr I ranges::remove(I first, S last, const T& value, Proj proj = {});
template<ForwardRange R, class T, class Proj = identity>
  requires Permutable<iterator_t<R>> &&
           IndirectRelation<ranges::equal_to, projected<iterator_t<R>, Proj>, const T*>
  constexpr safe_iterator_t<R>
    ranges::remove(R&& r, const T& value, Proj proj = {});
template<Permutable I, Sentinel<I> S, class Proj = identity,
         IndirectUnaryPredicate<projected<I, Proj>> Pred>
  constexpr I ranges::remove_if(I first, S last, Pred pred, Proj proj = {});
template<ForwardRange R, class Proj = identity,
         IndirectUnaryPredicate<projected<iterator_t<R>, Proj>> Pred>
  requires Permutable<iterator_t<R>>
  constexpr safe_iterator_t<R>
    ranges::remove_if(R&& r, Pred pred, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let $E$ be
\begin{itemize}
\item \tcode{bool(*i == value)} for \tcode{remove},
\item \tcode{bool(pred(*i))} for \tcode{remove_if},
\item \tcode{bool(invoke(proj, *i) == value)} for \tcode{ranges::remove}, or
\item \tcode{bool(invoke(pred, invoke(proj, *i)))} for \tcode{ranges::remove_if}.
\end{itemize}

\pnum
\requires
For the algorithms in namespace \tcode{std},
the type of \tcode{*first}
shall meet the \oldconcept{MoveAssignable} requirements (\tref{cpp17.moveassignable}).

\pnum
\effects
Eliminates all the elements referred to by iterator \tcode{i}
in the range \range{first}{last} for which $E$ holds.

\pnum
\returns
The end of the resulting range.

\pnum
\remarks
Stable\iref{algorithm.stable}.

\pnum
\complexity
Exactly \tcode{last - first} applications
of the corresponding predicate and any projection.

\pnum
\begin{note}
Each element in the range \range{ret}{last},
where \tcode{ret} is the returned value,
has a valid but unspecified state,
because the algorithms can eliminate elements
by moving from elements that were originally in that range.
\end{note}
\end{itemdescr}

\indexlibrary{\idxcode{remove_copy}}%
\indexlibrary{\idxcode{remove_copy_if}}%
\begin{itemdecl}
template<class InputIterator, class OutputIterator, class T>
  constexpr OutputIterator
    remove_copy(InputIterator first, InputIterator last,
                OutputIterator result, const T& value);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
         class T>
  ForwardIterator2
    remove_copy(ExecutionPolicy&& exec,
                ForwardIterator1 first, ForwardIterator1 last,
                ForwardIterator2 result, const T& value);

template<class InputIterator, class OutputIterator, class Predicate>
  constexpr OutputIterator
    remove_copy_if(InputIterator first, InputIterator last,
                   OutputIterator result, Predicate pred);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
         class Predicate>
  ForwardIterator2
    remove_copy_if(ExecutionPolicy&& exec,
                   ForwardIterator1 first, ForwardIterator1 last,
                   ForwardIterator2 result, Predicate pred);

template<InputIterator I, Sentinel<I> S, WeaklyIncrementable O, class T,
         class Proj = identity>
  requires IndirectlyCopyable<I, O> &&
           IndirectRelation<ranges::equal_to, projected<I, Proj>, const T*>
  constexpr ranges::remove_copy_result<I, O>
    ranges::remove_copy(I first, S last, O result, const T& value, Proj proj = {});
template<InputRange R, WeaklyIncrementable O, class T, class Proj = identity>
  requires IndirectlyCopyable<iterator_t<R>, O> &&
           IndirectRelation<ranges::equal_to, projected<iterator_t<R>, Proj>, const T*>
  constexpr ranges::remove_copy_result<safe_iterator_t<R>, O>
    ranges::remove_copy(R&& r, O result, const T& value, Proj proj = {});
template<InputIterator I, Sentinel<I> S, WeaklyIncrementable O,
         class Proj = identity, IndirectUnaryPredicate<projected<I, Proj>> Pred>
  requires IndirectlyCopyable<I, O>
  constexpr ranges::remove_copy_if_result<I, O>
    ranges::remove_copy_if(I first, S last, O result, Pred pred, Proj proj = {});
template<InputRange R, WeaklyIncrementable O, class Proj = identity,
         IndirectUnaryPredicate<projected<iterator_t<R>, Proj>> Pred>
  requires IndirectlyCopyable<iterator_t<R>, O>
  constexpr ranges::remove_copy_if_result<safe_iterator_t<R>, O>
    ranges::remove_copy_if(R&& r, O result, Pred pred, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let $E$ be
\begin{itemize}
\item \tcode{bool(*i == value)} for \tcode{remove_copy},
\item \tcode{bool(pred(*i))} for \tcode{remove_copy_if},
\item \tcode{bool(invoke(proj, *i) == value)} for \tcode{ranges::remove_copy}, or
\item \tcode{bool(invoke(pred, invoke(proj, *i)))} for \tcode{ranges::remove_copy_if}.
\end{itemize}

\pnum
Let $N$ be the number of elements in \range{first}{last}
for which $E$ is \tcode{false}.

\pnum
\requires
The ranges \range{first}{last} and \range{result}{result + (last - first)}
shall not overlap. The expression \tcode{*result = *first} shall be valid.
\begin{note}
For the overloads with an \tcode{ExecutionPolicy},
there may be a performance cost
if \tcode{iterator_traits<ForwardIterator1>::value_type} does not meet
the \oldconcept{\-Move\-Constructible} (\tref{cpp17.moveconstructible}) requirements.
\end{note}

\pnum
\effects
Copies all the elements referred to by the iterator \tcode{i}
in the range \range{first}{last} for which $E$ is \tcode{false}.

\pnum
\returns
\begin{itemize}
\item \tcode{result + $N$}, for the algorithms in namespace \tcode{std}, or
\item \tcode{\{last, result + $N$\}}, for the algorithms in namespace \tcode{ranges}.
\end{itemize}

\pnum
\complexity
Exactly \tcode{last - first} applications
of the corresponding predicate and any projection.

\pnum
\remarks
Stable\iref{algorithm.stable}.
\end{itemdescr}

\rSec2[alg.unique]{Unique}

\indexlibrary{\idxcode{unique}}%
\begin{itemdecl}
template<class ForwardIterator>
  constexpr ForwardIterator unique(ForwardIterator first, ForwardIterator last);
template<class ExecutionPolicy, class ForwardIterator>
  ForwardIterator unique(ExecutionPolicy&& exec,
                         ForwardIterator first, ForwardIterator last);

template<class ForwardIterator, class BinaryPredicate>
  constexpr ForwardIterator unique(ForwardIterator first, ForwardIterator last,
                                   BinaryPredicate pred);
template<class ExecutionPolicy, class ForwardIterator, class BinaryPredicate>
  ForwardIterator unique(ExecutionPolicy&& exec,
                         ForwardIterator first, ForwardIterator last,
                         BinaryPredicate pred);

template<Permutable I, Sentinel<I> S, class Proj = identity,
         IndirectRelation<projected<I, Proj>> C = ranges::equal_to>
  constexpr I ranges::unique(I first, S last, C comp = {}, Proj proj = {});
template<ForwardRange R, class Proj = identity,
         IndirectRelation<projected<iterator_t<R>, Proj>> C = ranges::equal_to>
  requires Permutable<iterator_t<R>>
  constexpr safe_iterator_t<R>
    ranges::unique(R&& r, C comp = {}, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{pred} be \tcode{equal_to\{\}}
for the overloads with no parameter \tcode{pred}, and
let $E$ be
\begin{itemize}
\setlength{\emergencystretch}{1em}
\item
  \tcode{bool(pred(*(i - 1), *i))}
  for the overloads in namespace \tcode{std}, or
\item
  \tcode{bool(invoke(comp, invoke(proj, *(i - 1)), invoke(proj, *i)))}
  for the overloads in namespace \tcode{ranges}.
\end{itemize}

\pnum
\requires
For the overloads in namepace \tcode{std},
\tcode{pred} shall be an equivalence relation and
the type of \tcode{*first} shall meet
the \oldconcept{MoveAssignable} requirements (\tref{cpp17.moveassignable}).

\pnum
\effects
For a nonempty range, eliminates all but the first element
from every consecutive group of equivalent elements referred to
by the iterator \tcode{i} in the range \range{first + 1}{last}
for which $E$ is \tcode{true}.

\pnum
\returns
The end of the resulting range.

\pnum
\complexity
For nonempty ranges, exactly \tcode{(last - first) - 1} applications
of the corresponding predicate and
no more than twice as many applications of any projection.
\end{itemdescr}

\indexlibrary{\idxcode{unique_copy}}%
\begin{itemdecl}
template<class InputIterator, class OutputIterator>
  constexpr OutputIterator
    unique_copy(InputIterator first, InputIterator last,
                OutputIterator result);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2>
  ForwardIterator2
    unique_copy(ExecutionPolicy&& exec,
                ForwardIterator1 first, ForwardIterator1 last,
                ForwardIterator2 result);

template<class InputIterator, class OutputIterator,
         class BinaryPredicate>
  constexpr OutputIterator
    unique_copy(InputIterator first, InputIterator last,
                OutputIterator result, BinaryPredicate pred);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
         class BinaryPredicate>
  ForwardIterator2
    unique_copy(ExecutionPolicy&& exec,
                ForwardIterator1 first, ForwardIterator1 last,
                ForwardIterator2 result, BinaryPredicate pred);

template<InputIterator I, Sentinel<I> S, WeaklyIncrementable O,
         class Proj = identity, IndirectRelation<projected<I, Proj>> C = ranges::equal_to>
  requires IndirectlyCopyable<I, O> &&
           (ForwardIterator<I> ||
            (InputIterator<O> && Same<iter_value_t<I>, iter_value_t<O>>) ||
            IndirectlyCopyableStorable<I, O>)
  constexpr ranges::unique_copy_result<I, O>
    ranges::unique_copy(I first, S last, O result, C comp = {}, Proj proj = {});
template<InputRange R, WeaklyIncrementable O, class Proj = identity,
         IndirectRelation<projected<iterator_t<R>, Proj>> C = ranges::equal_to>
  requires IndirectlyCopyable<iterator_t<R>, O> &&
           (ForwardIterator<iterator_t<R>> ||
            (InputIterator<O> && Same<iter_value_t<iterator_t<R>>, iter_value_t<O>>) ||
            IndirectlyCopyableStorable<iterator_t<R>, O>)
  constexpr ranges::unique_copy_result<safe_iterator_t<R>, O>
    ranges::unique_copy(R&& r, O result, C comp = {}, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{pred} be \tcode{equal_to\{\}} for the overloads
in namespace \tcode{std} with no parameter \tcode{pred}, and
let $E$ be
\begin{itemize}
\setlength{\emergencystretch}{1em}
\item
  \tcode{bool(pred(*i, *(i - 1)))}
  for the overloads in namespace \tcode{std}, or
\item
  \tcode{bool(invoke(comp, invoke(proj, *i), invoke(proj, *(i - 1))))}
  for the overloads in namespace \tcode{ranges}.
\end{itemize}

\pnum
\requires
\begin{itemize}
\item
  The ranges \range{first}{last} and \range{result}{result+(last-first)}
  shall not overlap.
\item
  For the overloads in namespace \tcode{std}:
  \begin{itemize}
  \item
    The comparison function shall be an equivalence relation.
  \item
    The expression \tcode{*result = *first} shall be valid.
  \item
    For the overloads with no \tcode{ExecutionPolicy},
    let \tcode{T} be the value type of \tcode{InputIterator}.
    If \tcode{InputIterator} meets
    the \oldconcept{ForwardIterator} requirements,
    then there are no additional requirements for \tcode{T}.
    Otherwise, if \tcode{OutputIterator} meets
    the \oldconcept{ForwardIterator} requirements and
    its value type is the same as \tcode{T},
    then \tcode{T} shall meet
    the \oldconcept{CopyAssignable} (\tref{cpp17.copyassignable}) requirements.
    Otherwise, \tcode{T} shall meet both
    the \oldconcept{CopyConstructible} (\tref{cpp17.copyconstructible}) and
    \oldconcept{CopyAssignable} requirements.
    \begin{note}
    For the overloads with an \tcode{ExecutionPolicy},
    there may be a performance cost
    if the value type of \tcode{ForwardIterator1} does not meet both the
    \oldconcept{CopyConstructible} and \oldconcept{CopyAssignable} requirements.
    \end{note}
  \end{itemize}
\end{itemize}

\pnum
\effects
Copies only the first element from every consecutive group of equal elements
referred to by the iterator \tcode{i} in the range \range{first}{last}
for which $E$ holds.

\pnum
\returns
\begin{itemize}
\item \tcode{result + $N$} for the overloads in namespace \tcode{std}, or
\item \tcode{\{last, result + $N$\}} for the overloads in namespace \tcode{ranges}
\end{itemize}

\pnum
\complexity
Exactly \tcode{last - first - 1} applications
of the corresponding predicate
and no more than twice as many applications of any projection.
\end{itemdescr}

\rSec2[alg.reverse]{Reverse}

\indexlibrary{\idxcode{reverse}}%
\begin{itemdecl}
template<class BidirectionalIterator>
  constexpr void reverse(BidirectionalIterator first, BidirectionalIterator last);
template<class ExecutionPolicy, class BidirectionalIterator>
  void reverse(ExecutionPolicy&& exec,
               BidirectionalIterator first, BidirectionalIterator last);

template<BidirectionalIterator I, Sentinel<I> S>
  requires Permutable<I>
  constexpr I ranges::reverse(I first, S last);
template<BidirectionalRange R>
  requires Permutable<iterator_t<R>>
  constexpr safe_iterator_t<R> ranges::reverse(R&& r);
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
For the overloads in namespace \tcode{std},
\tcode{BidirectionalIterator} shall meet
the \oldconcept{Value\-Swappable} requirements\iref{swappable.requirements}.

\pnum
\effects
For each non-negative integer \tcode{i < (last - first) / 2},
applies \tcode{std::iter_swap}, or
\tcode{ranges::\brk{}iter_swap} for the overloads in namespace \tcode{ranges},
to all pairs of iterators \tcode{first + i, (last - i) - 1}.

\pnum
\returns
\tcode{last} for the overloads in namespace \tcode{ranges}.

\pnum
\complexity
Exactly \tcode{(last - first)/2} swaps.
\end{itemdescr}

\indexlibrary{\idxcode{reverse_copy}}%
\begin{itemdecl}
template<class BidirectionalIterator, class OutputIterator>
  constexpr OutputIterator
    reverse_copy(BidirectionalIterator first, BidirectionalIterator last,
                 OutputIterator result);
template<class ExecutionPolicy, class BidirectionalIterator, class ForwardIterator>
  ForwardIterator
    reverse_copy(ExecutionPolicy&& exec,
                 BidirectionalIterator first, BidirectionalIterator last,
                 ForwardIterator result);

template<BidirectionalIterator I, Sentinel<I> S, WeaklyIncrementable O>
  requires IndirectlyCopyable<I, O>
  constexpr ranges::reverse_copy_result<I, O>
    ranges::reverse_copy(I first, S last, O result);
template<BidirectionalRange R, WeaklyIncrementable O>
  requires IndirectlyCopyable<iterator_t<R>, O>
  constexpr ranges::reverse_copy_result<safe_iterator_t<R>, O>
    ranges::reverse_copy(R&& r, O result);
\end{itemdecl}

\begin{itemdescr}
\pnum
Let $N$ be \tcode{last - first}.

\pnum
\requires
The ranges \range{first}{last} and \range{result}{result + $N$}
shall not overlap.

\pnum
\effects
Copies the range \range{first}{last} to the range \range{result}{result + $N$}
such that for every non-negative integer \tcode{i < $N$}
the following assignment takes place:
\tcode{*(result + $N$ - 1 - i) = *(first + i)}.

\pnum
\returns
\begin{itemize}
\item
  \tcode{result + $N$} for the overloads in namespace \tcode{std}, or
\item
  \tcode{\{last, result + $N$\}} for the overloads in namespace \tcode{ranges}.
\end{itemize}

\pnum
\complexity
Exactly $N$ assignments.
\end{itemdescr}

\rSec2[alg.rotate]{Rotate}

\indexlibrary{\idxcode{rotate}}%
\begin{itemdecl}
template<class ForwardIterator>
  constexpr ForwardIterator
    rotate(ForwardIterator first, ForwardIterator middle, ForwardIterator last);
template<class ExecutionPolicy, class ForwardIterator>
  ForwardIterator
    rotate(ExecutionPolicy&& exec,
           ForwardIterator first, ForwardIterator middle, ForwardIterator last);

template<Permutable I, Sentinel<I> S>
  constexpr subrange<I> ranges::rotate(I first, I middle, S last);
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
\range{first}{middle} and \range{middle}{last} shall be valid ranges.
For the overloads in namespace \tcode{std},
\tcode{ForwardIterator} shall meet
the \oldconcept{ValueSwappable} requirements\iref{swappable.requirements}, and
the type of \tcode{*first} shall meet
the \oldconcept{MoveConstructible} (\tref{cpp17.moveconstructible}) and
\oldconcept{MoveAssignable} (\tref{cpp17.moveassignable}) requirements.

\pnum
\effects
For each non-negative integer \tcode{i < (last - first)},
places the element from the position \tcode{first + i}
into position \tcode{first + (i + (last - middle)) \% (last - first)}.
\begin{note}
This is a left rotate.
\end{note}

\pnum
\returns
\begin{itemize}
\item
  \tcode{first + (last - middle)}
  for the overloads in namespace \tcode{std}, or
\item
  \tcode{\{first + (last - middle), last\}}
  for the overload in namespace \tcode{ranges}.
\end{itemize}

\pnum
\complexity
At most \tcode{last - first} swaps.
\end{itemdescr}

\begin{itemdecl}
template<ForwardRange R>
  requires Permutable<iterator_t<R>>
  constexpr safe_subrange_t<R> ranges::rotate(R&& r, iterator_t<R> middle);
\end{itemdecl}

\begin{itemdescr}
\pnum
\effects
Equivalent to:
\tcode{return ranges::rotate(ranges::begin(r), middle, ranges::end(r));}
\end{itemdescr}

\indexlibrary{\idxcode{rotate_copy}}%
\begin{itemdecl}
template<class ForwardIterator, class OutputIterator>
  constexpr OutputIterator
    rotate_copy(ForwardIterator first, ForwardIterator middle, ForwardIterator last,
                OutputIterator result);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2>
  ForwardIterator2
    rotate_copy(ExecutionPolicy&& exec,
                ForwardIterator1 first, ForwardIterator1 middle, ForwardIterator1 last,
                ForwardIterator2 result);

  template<ForwardIterator I, Sentinel<I> S, WeaklyIncrementable O>
    requires IndirectlyCopyable<I, O>
    constexpr ranges::rotate_copy_result<I, O>
      ranges::rotate_copy(I first, I middle, S last, O result);
\end{itemdecl}

\begin{itemdescr}
\pnum
Let $N$ be \tcode{last - first}.

\pnum
\requires
\range{first}{middle} and \range{middle}{last} shall be valid ranges.
The ranges \range{first}{last} and \range{result}{result + $N$}
shall not overlap.

\pnum
\effects
Copies the range \range{first}{last} to the range \range{result}{result + $N$}
such that for each non-negative integer $i < N$
the following assignment takes place:
\tcode{*(result + $i$) =  *(first + ($i$ + (middle - first)) \% $N$)}.

\pnum
\returns
\begin{itemize}
\item
  \tcode{result + $N$} for the overloads in namespace \tcode{std}, or
\item
  \tcode{\{last, result + $N$\}} for the overload in namespace \tcode{ranges}.
\end{itemize}

\pnum
\complexity
Exactly $N$ assignments.
\end{itemdescr}

\begin{itemdecl}
template<ForwardRange R, WeaklyIncrementable O>
  requires IndirectlyCopyable<iterator_t<R>, O>
  constexpr ranges::rotate_copy_result<safe_iterator_t<R>, O>
    ranges::rotate_copy(R&& r, iterator_t<R> middle, O result);
\end{itemdecl}

\begin{itemdescr}
\pnum
\effects
Equivalent to:
\begin{codeblock}
return ranges::rotate_copy(ranges::begin(r), middle, ranges::end(r), result);
\end{codeblock}
\end{itemdescr}

\rSec2[alg.random.sample]{Sample}

\indexlibrary{\idxcode{sample}}%
\begin{itemdecl}
template<class PopulationIterator, class SampleIterator,
         class Distance, class UniformRandomBitGenerator>
  SampleIterator sample(PopulationIterator first, PopulationIterator last,
                        SampleIterator out, Distance n,
                        UniformRandomBitGenerator&& g);
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
\begin{itemize}
\item
  \tcode{PopulationIterator} shall satisfy
  the \oldconcept{InputIterator} requirements\iref{input.iterators}.
\item
  \tcode{SampleIterator} shall satisfy
  the \oldconcept{OutputIterator} requirements\iref{output.iterators}.
\item
  \tcode{SampleIterator} shall satisfy
  the \oldconcept{RandomAccessIterator} requirements\iref{random.access.iterators}
  unless \tcode{Pop\-ulat\-ion\-Iter\-ator} satisfies
  the \oldconcept{ForwardIterator} requirements\iref{forward.iterators}.
\item
  \tcode{PopulationIterator}'s value type shall be
  writable\iref{iterator.requirements.general} to \tcode{out}.
\item
  \tcode{Distance} shall be an integer type.
\item
  \tcode{remove_reference_t<UniformRandomBitGenerator>} shall satisfy
  the requirements of a uniform random bit generator type\iref{rand.req.urng}
  whose return type is convertible to \tcode{Distance}.
\item
  \tcode{out} shall not be in the range \range{first}{last}.
\end{itemize}

\pnum
\effects
Copies $\min(\tcode{last - first}, \ \tcode{n})$ elements (the \defn{sample})
from \range{first}{last} (the \defn{population}) to \tcode{out}
such that each possible sample has equal probability of appearance.
\begin{note}
Algorithms that obtain such effects include \term{selection sampling}
and \term{reservoir sampling}.
\end{note}

\pnum
\returns
The end of the resulting sample range.

\pnum
\complexity
\bigoh{\tcode{last - first}}.

\pnum
\remarks
\begin{itemize}
\item
  Stable if and only if \tcode{PopulationIterator} satisfies
  the \oldconcept{ForwardIterator} requirements.
\item
  To the extent that the implementation of this function makes use
  of random numbers, the object \tcode{g} shall serve as
  the implementation's source of randomness.
\end{itemize}
\end{itemdescr}

\rSec2[alg.random.shuffle]{Shuffle}

\indexlibrary{\idxcode{shuffle}}%
\begin{itemdecl}
template<class RandomAccessIterator, class UniformRandomBitGenerator>
  void shuffle(RandomAccessIterator first,
               RandomAccessIterator last,
               UniformRandomBitGenerator&& g);

template<RandomAccessIterator I, Sentinel<I> S, class Gen>
  requires Permutable<I> &&
           UniformRandomBitGenerator<remove_reference_t<Gen>>
  I ranges::shuffle(I first, S last, Gen&& g);
template<RandomAccessRange R, class Gen>
  requires Permutable<iterator_t<R>> &&
           UniformRandomBitGenerator<remove_reference_t<Gen>>
  safe_iterator_t<R> ranges::shuffle(R&& r, Gen&& g);
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
For the overload in namespace \tcode{std}:
\begin{itemize}
\item
  \tcode{RandomAccessIterator} shall meet
  the \oldconcept{ValueSwappable} requirements\iref{swappable.requirements}.
\item
  The type \tcode{remove_reference_t<UniformRandomBitGenerator>} shall meet
  the uniform random bit generator\iref{rand.req.urng} requirements.
\end{itemize}

\pnum
\effects
Permutes the elements in the range \range{first}{last}
such that each possible permutation of those elements
has equal probability of appearance.

\pnum
\returns
\tcode{last} for the overloads in namespace \tcode{ranges}.

\pnum
\complexity
Exactly \tcode{(last - first) - 1} swaps.

\pnum
\remarks
To the extent that the implementation of this function makes use
of random numbers, the object referenced by \tcode{g} shall serve as
the implementation's source of randomness.
\end{itemdescr}

\rSec2[alg.shift]{Shift}

\indexlibrary{\idxcode{shift_left}}%
\begin{itemdecl}
template<class ForwardIterator>
  constexpr ForwardIterator
    shift_left(ForwardIterator first, ForwardIterator last,
               typename iterator_traits<ForwardIterator>::difference_type n);
template<class ExecutionPolicy, class ForwardIterator>
  ForwardIterator
    shift_left(ExecutionPolicy&& exec, ForwardIterator first, ForwardIterator last,
               typename iterator_traits<ForwardIterator>::difference_type n);
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
The type of \tcode{*first} shall satisfy
the \oldconcept{MoveAssignable} requirements.

\pnum
\effects
If \tcode{n <= 0} or \tcode{n >= last - first}, does nothing.
Otherwise, moves the element
from position \tcode{first + n + i}
into position \tcode{first + i}
for each non-negative integer \tcode{i < (last - first) - n}.
In the first overload case, does so in order starting
from \tcode{i = 0} and proceeding to \tcode{i = (last - first) - n - 1}.

\pnum
\returns
\tcode{first + (last - first - n)}
if \tcode{n} is positive and \tcode{n < last - first},
otherwise \tcode{first} if \tcode{n} is positive, otherwise \tcode{last}.

\pnum
\complexity
At most \tcode{(last - first) - n} assignments.
\end{itemdescr}

\indexlibrary{\idxcode{shift_right}}%
\begin{itemdecl}
template<class ForwardIterator>
  constexpr ForwardIterator
    shift_right(ForwardIterator first, ForwardIterator last,
                typename iterator_traits<ForwardIterator>::difference_type n);
template<class ExecutionPolicy, class ForwardIterator>
  ForwardIterator
    shift_right(ExecutionPolicy&& exec, ForwardIterator first, ForwardIterator last,
                typename iterator_traits<ForwardIterator>::difference_type n);
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
The type of \tcode{*first} shall satisfy
the \oldconcept{MoveAssignable} requirements.
\tcode{ForwardIterator} shall meet
the \oldconcept{BidirectionalIterator} requirements\iref{bidirectional.iterators} or
the \oldconcept{ValueSwappable} requirements.

\pnum
\effects
If \tcode{n <= 0} or \tcode{n >= last - first}, does nothing.
Otherwise, moves the element
from position \tcode{first + i} into \tcode{position first + n + i}
for each non-negative integer \tcode{i < (last - first) - n}.
In the first overload case, if \tcode{ForwardIterator} satisfies
the \oldconcept{BidirectionalIterator} requirements,
does so in order starting
from \tcode{i = (last - first) - n - 1} and proceeding to \tcode{i = 0}.

\pnum
\returns
\tcode{first + n}
if \tcode{n} is positive and \tcode{n < last - first},
otherwise \tcode{last} if \tcode{n} is positive, otherwise \tcode{first}.

\pnum
\complexity
At most \tcode{(last - first) - n} assignments or swaps.
\end{itemdescr}

\rSec1[alg.sorting]{Sorting and related operations}

\pnum
The operations in~\ref{alg.sorting} defined directly in namespace \tcode{std}
have two versions:
one that takes a function object of type \tcode{Compare} and
one that uses an \tcode{operator<}.

\pnum
\tcode{Compare} is a function object type\iref{function.objects}
that meets the requirements for a template parameter
named \tcode{BinaryPredicate}~\iref{algorithms.requirements}.
The return value of the function call operation
applied to an object of type \tcode{Compare},
when contextually converted to \tcode{bool}\iref{conv},
yields \tcode{true}
if the first argument of the call is less than the second, and
\tcode{false} otherwise.
\tcode{Compare comp} is used throughout
for algorithms assuming an ordering relation.

\pnum
For all algorithms that take \tcode{Compare},
there is a version that uses \tcode{operator<} instead.
That is, \tcode{comp(*i, *j) != false} defaults to \tcode{*i < *j != false}.
For algorithms other than those described in~\ref{alg.binary.search},
\tcode{comp} shall induce a strict weak ordering on the values.

\pnum
The term \term{strict} refers to the requirement
of an irreflexive relation (\tcode{!comp(x, x)} for all \tcode{x}),
and the term \term{weak} to requirements
that are not as strong as those for a total ordering,
but stronger than those for a partial ordering.
If we define \tcode{equiv(a, b)} as \tcode{!comp(a, b) \&\& !comp(b, a)},
then the requirements are that \tcode{comp} and \tcode{equiv}
both be transitive relations:

\begin{itemize}
\item \tcode{comp(a, b) \&\& comp(b, c)} implies \tcode{comp(a, c)}
\item \tcode{equiv(a, b) \&\& equiv(b, c)} implies \tcode{equiv(a, c)}
\end{itemize}
\begin{note}
Under these conditions, it can be shown that
\begin{itemize}
\item
  \tcode{equiv} is an equivalence relation,
\item
  \tcode{comp} induces a well-defined relation
  on the equivalence classes determined by \tcode{equiv}, and
\item
  the induced relation is a strict total ordering.
\end{itemize}
\end{note}

\pnum
A sequence is \term{sorted with respect to a \tcode{comp} and \tcode{proj}}
for a comparator and projection \tcode{comp} and \tcode{proj}
if for every iterator \tcode{i} pointing to the sequence and
every non-negative integer \tcode{n}
such that \tcode{i + n} is a valid iterator
pointing to an element of the sequence,
\begin{codeblock}
bool(invoke(comp, invoke(proj, *(i + n)), invoke(proj, *i)))
\end{codeblock}
is \tcode{false}.

\pnum
A sequence \range{start}{finish} is
\term{partitioned with respect to an expression} \tcode{f(e)}
if there exists an integer \tcode{n}
such that for all \tcode{0 <= i < (finish - start)},
\tcode{f(*(start + i))} is \tcode{true} if and only if \tcode{i < n}.

\pnum
In the descriptions of the functions that deal with ordering relationships
we frequently use a notion of equivalence to describe concepts
such as stability.
The equivalence to which we refer is not necessarily an \tcode{operator==},
but an equivalence relation induced by the strict weak ordering.
That is, two elements \tcode{a} and \tcode{b} are considered equivalent
if and only if \tcode{!(a < b) \&\& !(b < a)}.

\rSec2[alg.sort]{Sorting}

\rSec3[sort]{\tcode{sort}}

\indexlibrary{\idxcode{sort}}%
\begin{itemdecl}
template<class RandomAccessIterator>
  constexpr void sort(RandomAccessIterator first, RandomAccessIterator last);
template<class ExecutionPolicy, class RandomAccessIterator>
  void sort(ExecutionPolicy&& exec,
            RandomAccessIterator first, RandomAccessIterator last);

template<class RandomAccessIterator, class Compare>
  constexpr void sort(RandomAccessIterator first, RandomAccessIterator last,
                      Compare comp);
template<class ExecutionPolicy, class RandomAccessIterator, class Compare>
  void sort(ExecutionPolicy&& exec,
            RandomAccessIterator first, RandomAccessIterator last,
            Compare comp);

template<RandomAccessIterator I, Sentinel<I> S, class Comp = ranges::less,
         class Proj = identity>
  requires Sortable<I, Comp, Proj>
  constexpr I
    ranges::sort(I first, S last, Comp comp = {}, Proj proj = {});
template<RandomAccessRange R, class Comp = ranges::less, class Proj = identity>
  requires Sortable<iterator_t<R>, Comp, Proj>
  constexpr safe_iterator_t<R>
    ranges::sort(R&& r, Comp comp = {}, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{comp} be \tcode{less\{\}}
and \tcode{proj} be \tcode{identity\{\}}
for the overloads with no parameters by those names.

\pnum
\requires
For the overloads in namespace \tcode{std},
\tcode{RandomAccessIterator} shall meet
the \oldconcept{Value\-Swappable} requirements\iref{swappable.requirements} and
the type of \tcode{*first} shall meet
the \oldconcept{MoveConstructible} (\tref{cpp17.moveconstructible}) and
\oldconcept{MoveAssignable} (\tref{cpp17.moveassignable}) requirements.

\pnum
\effects
Sorts the elements in the range \range{first}{last}
with respect to \tcode{comp} and \tcode{proj}.

\pnum
\returns
\tcode{last}, for the overloads in namespace \tcode{ranges}.

\pnum
\complexity
Let $N$ be \tcode{last - first}.
\bigoh{N \log N} comparisons and projections.
\end{itemdescr}

\rSec3[stable.sort]{\tcode{stable_sort}}

\indexlibrary{\idxcode{stable_sort}}%
\begin{itemdecl}
template<class RandomAccessIterator>
  void stable_sort(RandomAccessIterator first, RandomAccessIterator last);
template<class ExecutionPolicy, class RandomAccessIterator>
  void stable_sort(ExecutionPolicy&& exec,
                   RandomAccessIterator first, RandomAccessIterator last);

template<class RandomAccessIterator, class Compare>
  void stable_sort(RandomAccessIterator first, RandomAccessIterator last,
                   Compare comp);
template<class ExecutionPolicy, class RandomAccessIterator, class Compare>
  void stable_sort(ExecutionPolicy&& exec,
                   RandomAccessIterator first, RandomAccessIterator last,
                   Compare comp);

template<RandomAccessIterator I, Sentinel<I> S, class Comp = ranges::less,
         class Proj = identity>
  requires Sortable<I, Comp, Proj>
  I ranges::stable_sort(I first, S last, Comp comp = {}, Proj proj = {});
template<RandomAccessRange R, class Comp = ranges::less, class Proj = identity>
  requires Sortable<iterator_t<R>, Comp, Proj>
  safe_iterator_t<R>
    ranges::stable_sort(R&& r, Comp comp = {}, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{comp} be \tcode{less\{\}}
and \tcode{proj} be \tcode{identity\{\}}
for the overloads with no parameters by those names.

\pnum
\requires
For the overloads in namespace \tcode{std},
\tcode{RandomAccessIterator} shall meet
the \oldconcept{Value\-Swappable} requirements\iref{swappable.requirements} and
the type of \tcode{*first} shall meet
the \oldconcept{MoveConstructible} (\tref{cpp17.moveconstructible}) and
\oldconcept{MoveAssignable} (\tref{cpp17.moveassignable}) requirements.

\pnum
\effects
Sorts the elements in the range \range{first}{last}
with respect to \tcode{comp} and \tcode{proj}.

\pnum
\returns
\tcode{last} for the overloads in namespace \tcode{ranges}.

\pnum
\complexity
Let $N$ be \tcode{last - first}.
If enough extra memory is available, $N \log(N)$ comparisons.
Otherwise, at most $N \log^2(N)$ comparisons.
In either case, twice as many projections as the number of comparisons.

\pnum
\remarks
Stable\iref{algorithm.stable}.
\end{itemdescr}

\rSec3[partial.sort]{\tcode{partial_sort}}

\indexlibrary{\idxcode{partial_sort}}%
\begin{itemdecl}
template<class RandomAccessIterator>
  constexpr void partial_sort(RandomAccessIterator first,
                              RandomAccessIterator middle,
                              RandomAccessIterator last);
template<class ExecutionPolicy, class RandomAccessIterator>
  void partial_sort(ExecutionPolicy&& exec,
                    RandomAccessIterator first,
                    RandomAccessIterator middle,
                    RandomAccessIterator last);

template<class RandomAccessIterator, class Compare>
  constexpr void partial_sort(RandomAccessIterator first,
                              RandomAccessIterator middle,
                              RandomAccessIterator last,
                              Compare comp);
template<class ExecutionPolicy, class RandomAccessIterator, class Compare>
  void partial_sort(ExecutionPolicy&& exec,
                    RandomAccessIterator first,
                    RandomAccessIterator middle,
                    RandomAccessIterator last,
                    Compare comp);

template<RandomAccessIterator I, Sentinel<I> S, class Comp = ranges::less,
         class Proj = identity>
  requires Sortable<I, Comp, Proj>
  constexpr I
    ranges::partial_sort(I first, I middle, S last, Comp comp = {}, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{comp} be \tcode{less\{\}}
and \tcode{proj} be \tcode{identity\{\}}
for the overloads with no parameters by those names.

\pnum
\requires
\range{first}{middle} and \range{middle}{last} shall be valid ranges.
For the overloads in namespace \tcode{std},
\tcode{RandomAccessIterator} shall meet
the \oldconcept{ValueSwappable} requirements\iref{swappable.requirements} and
the type of \tcode{*first} shall meet
the \oldconcept{MoveConstructible} (\tref{cpp17.moveconstructible}) and
\oldconcept{MoveAssignable} (\tref{cpp17.moveassignable}) requirements.

\pnum
\effects
Places the first \tcode{middle - first} elements
from the range \range{first}{last}
as sorted with respect to \tcode{comp} and \tcode{proj}
into the range \range{first}{middle}.
The rest of the elements in the range \range{middle}{last}
are placed in an unspecified order.
\indextext{unspecified}%

\pnum
\returns
\tcode{last} for the overload in namespace \tcode{ranges}.

\pnum
\complexity
Approximately \tcode{(last - first) * log(middle - first)} comparisons, and
twice as many projections.
\end{itemdescr}

\begin{itemdecl}
template<RandomAccessRange R, class Comp = ranges::less, class Proj = identity>
  requires Sortable<iterator_t<R>, Comp, Proj>
  constexpr safe_iterator_t<R>
    ranges::partial_sort(R&& r, iterator_t<R> middle, Comp comp = {}, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
\effects
Equivalent to:
\begin{codeblock}
return ranges::partial_sort(ranges::begin(r), middle, ranges::end(r), comp, proj);
\end{codeblock}
\end{itemdescr}

\rSec3[partial.sort.copy]{\tcode{partial_sort_copy}}

\indexlibrary{\idxcode{partial_sort_copy}}%
\begin{itemdecl}
template<class InputIterator, class RandomAccessIterator>
  constexpr RandomAccessIterator
    partial_sort_copy(InputIterator first, InputIterator last,
                      RandomAccessIterator result_first,
                      RandomAccessIterator result_last);
template<class ExecutionPolicy, class ForwardIterator, class RandomAccessIterator>
  RandomAccessIterator
    partial_sort_copy(ExecutionPolicy&& exec,
                      ForwardIterator first, ForwardIterator last,
                      RandomAccessIterator result_first,
                      RandomAccessIterator result_last);

template<class InputIterator, class RandomAccessIterator,
         class Compare>
  constexpr RandomAccessIterator
    partial_sort_copy(InputIterator first, InputIterator last,
                      RandomAccessIterator result_first,
                      RandomAccessIterator result_last,
                      Compare comp);
template<class ExecutionPolicy, class ForwardIterator, class RandomAccessIterator,
         class Compare>
  RandomAccessIterator
    partial_sort_copy(ExecutionPolicy&& exec,
                      ForwardIterator first, ForwardIterator last,
                      RandomAccessIterator result_first,
                      RandomAccessIterator result_last,
                      Compare comp);

template<InputIterator I1, Sentinel<I1> S1, RandomAccessIterator I2, Sentinel<I2> S2,
         class Comp = ranges::less, class Proj1 = identity, class Proj2 = identity>
  requires IndirectlyCopyable<I1, I2> && Sortable<I2, Comp, Proj2> &&
           IndirectStrictWeakOrder<Comp, projected<I1, Proj1>, projected<I2, Proj2>>
  constexpr I2
    ranges::partial_sort_copy(I1 first, S1 last, I2 result_first, S2 result_last,
                              Comp comp = {}, Proj1 proj1 = {}, Proj2 proj2 = {});
template<InputRange R1, RandomAccessRange R2, class Comp = ranges::less,
         class Proj1 = identity, class Proj2 = identity>
  requires IndirectlyCopyable<iterator_t<R1>, iterator_t<R2>> &&
           Sortable<iterator_t<R2>, Comp, Proj2> &&
           IndirectStrictWeakOrder<Comp, projected<iterator_t<R1>, Proj1>,
                                   projected<iterator_t<R2>, Proj2>>
  constexpr safe_iterator_t<R2>
    ranges::partial_sort_copy(R1&& r, R2&& result_r, Comp comp = {},
                              Proj1 proj1 = {}, Proj2 proj2 = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let $N$ be $\min(\tcode{last - first}, \ \tcode{result_last - result_first})$.
Let \tcode{comp} be \tcode{less\{\}}, and
\tcode{proj1} and \tcode{proj2} be \tcode{identity\{\}}
for the overloads with no parameters by those names.

\pnum
\requires
For the overloads in namespace \tcode{std},
\tcode{RandomAccessIterator} shall meet
the \oldconcept{Value\-Swappable} requirements\iref{swappable.requirements},
the type of \tcode{*result_first} shall meet
the \oldconcept{MoveConstructible} (\tref{cpp17.moveconstructible}) and
\oldconcept{\-Move\-Assignable} (\tref{cpp17.moveassignable}) requirements,
and the expression \tcode{*first}
shall be writable\iref{iterator.requirements.general} to \tcode{result_first}.

\pnum
\expects
For iterators \tcode{a1} and \tcode{b1} in \range{first}{last}, and
iterators \tcode{x2} and \tcode{y2} in \range{result_first}{result_last},
after evaluating the assignment \tcode{*y2 = *b1}, let $E$ be the value of
\begin{codeblock}
bool(invoke(comp, invoke(proj1, *a1), invoke(proj2, *y2))).
\end{codeblock}
Then, after evaluating the assignment \tcode{*x2 = *a1}, $E$ is equal to
\begin{codeblock}
bool(invoke(comp, invoke(proj2, *x2), invoke(proj2, *y2))).
\end{codeblock}
\begin{note}
Writing a value from the input range into the output range does not affect
how it is ordered by \tcode{comp} and \tcode{proj1} or \tcode{proj2}.
\end{note}

\pnum
\effects
Places the first $N$ elements
as sorted with respect to \tcode{comp} and \tcode{proj2}
into the range \range{result_first}{result_first + $N$}.

\pnum
\returns
\tcode{result_first + $N$}.

\pnum
\complexity
Approximately \tcode{(last - first) * log $N$} comparisons,
and twice as many projections.
\end{itemdescr}

\rSec3[is.sorted]{\tcode{is_sorted}}

\indexlibrary{\idxcode{is_sorted}}%
\begin{itemdecl}
template<class ForwardIterator>
  constexpr bool is_sorted(ForwardIterator first, ForwardIterator last);
\end{itemdecl}

\begin{itemdescr}
\pnum
\effects
Equivalent to: \tcode{return is_sorted_until(first, last) == last;}
\end{itemdescr}

\indexlibrary{\idxcode{is_sorted}}%
\begin{itemdecl}
template<class ExecutionPolicy, class ForwardIterator>
  bool is_sorted(ExecutionPolicy&& exec,
                 ForwardIterator first, ForwardIterator last);
\end{itemdecl}

\begin{itemdescr}
\pnum
\effects
Equivalent to:
\begin{codeblock}
return is_sorted_until(std::forward<ExecutionPolicy>(exec), first, last) == last;
\end{codeblock}
\end{itemdescr}

\indexlibrary{\idxcode{is_sorted}}%
\begin{itemdecl}
template<class ForwardIterator, class Compare>
  constexpr bool is_sorted(ForwardIterator first, ForwardIterator last,
                           Compare comp);
\end{itemdecl}

\begin{itemdescr}
\pnum
\effects
Equivalent to: \tcode{return is_sorted_until(first, last, comp) == last;}
\end{itemdescr}


\indexlibrary{\idxcode{is_sorted}}%
\begin{itemdecl}
template<class ExecutionPolicy, class ForwardIterator, class Compare>
  bool is_sorted(ExecutionPolicy&& exec,
                 ForwardIterator first, ForwardIterator last,
                 Compare comp);
\end{itemdecl}

\begin{itemdescr}
\pnum
\effects
Equivalent to:
\begin{codeblock}
is_sorted_until(std::forward<ExecutionPolicy>(exec), first, last, comp) == last
\end{codeblock}
\end{itemdescr}

\indexlibrary{\idxcode{is_sorted}}%
\begin{itemdecl}
template<ForwardIterator I, Sentinel<I> S, class Proj = identity,
         IndirectStrictWeakOrder<projected<I, Proj>> Comp = ranges::less>
  constexpr bool ranges::is_sorted(I first, S last, Comp comp = {}, Proj proj = {});
template<ForwardRange R, class Proj = identity,
         IndirectStrictWeakOrder<projected<iterator_t<R>, Proj>> Comp = ranges::less>
  constexpr bool ranges::is_sorted(R&& r, Comp comp = {}, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
\effects
Equivalent to:
\tcode{return ranges::is_sorted_until(first, last, comp, proj) == last;}
\end{itemdescr}

\indexlibrary{\idxcode{is_sorted_until}}%
\begin{itemdecl}
template<class ForwardIterator>
  constexpr ForwardIterator
    is_sorted_until(ForwardIterator first, ForwardIterator last);
template<class ExecutionPolicy, class ForwardIterator>
  ForwardIterator
    is_sorted_until(ExecutionPolicy&& exec,
                    ForwardIterator first, ForwardIterator last);

template<class ForwardIterator, class Compare>
  constexpr ForwardIterator
    is_sorted_until(ForwardIterator first, ForwardIterator last,
                    Compare comp);
template<class ExecutionPolicy, class ForwardIterator, class Compare>
  ForwardIterator
    is_sorted_until(ExecutionPolicy&& exec,
                    ForwardIterator first, ForwardIterator last,
                    Compare comp);

template<ForwardIterator I, Sentinel<I> S, class Proj = identity,
         IndirectStrictWeakOrder<projected<I, Proj>> Comp = ranges::less>
  constexpr I ranges::is_sorted_until(I first, S last, Comp comp = {}, Proj proj = {});
template<ForwardRange R, class Proj = identity,
         IndirectStrictWeakOrder<projected<iterator_t<R>, Proj>> Comp = ranges::less>
  constexpr safe_iterator_t<R>
    ranges::is_sorted_until(R&& r, Comp comp = {}, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{comp} be \tcode{less\{\}}
and \tcode{proj} be \tcode{identity\{\}}
for the overloads with no parameters by those names.

\pnum
\returns
The last iterator \tcode{i} in \crange{first}{last}
for which the range \range{first}{i}
is sorted with respect to \tcode{comp} and \tcode{proj}.

\pnum
\complexity
Linear.
\end{itemdescr}

\rSec2[alg.nth.element]{Nth element}

\indexlibrary{\idxcode{nth_element}}%
\begin{itemdecl}
template<class RandomAccessIterator>
  constexpr void nth_element(RandomAccessIterator first, RandomAccessIterator nth,
                             RandomAccessIterator last);
template<class ExecutionPolicy, class RandomAccessIterator>
  void nth_element(ExecutionPolicy&& exec,
                   RandomAccessIterator first, RandomAccessIterator nth,
                   RandomAccessIterator last);

template<class RandomAccessIterator, class Compare>
  constexpr void nth_element(RandomAccessIterator first, RandomAccessIterator nth,
                             RandomAccessIterator last,  Compare comp);
template<class ExecutionPolicy, class RandomAccessIterator, class Compare>
  void nth_element(ExecutionPolicy&& exec,
                   RandomAccessIterator first, RandomAccessIterator nth,
                   RandomAccessIterator last, Compare comp);

template<RandomAccessIterator I, Sentinel<I> S, class Comp = ranges::less,
         class Proj = identity>
  requires Sortable<I, Comp, Proj>
  constexpr I
    ranges::nth_element(I first, I nth, S last, Comp comp = {}, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{comp} be \tcode{less\{\}}
and \tcode{proj} be \tcode{identity\{\}}
for the overloads with no parameters by those names.

\pnum
\requires
\range{first}{nth} and \range{nth}{last} shall be valid ranges.
For the overloads in namespace \tcode{std},
\tcode{RandomAccessIterator} shall meet
the \oldconcept{ValueSwappable} requirements\iref{swappable.requirements}, and
the type of \tcode{*first} shall meet
the \oldconcept{MoveConstructible} (\tref{cpp17.moveconstructible}) and
\oldconcept{MoveAssignable} (\tref{cpp17.moveassignable}) requirements.

\pnum
\effects
After \tcode{nth_element} the element in the position pointed to by \tcode{nth}
is the element that would be in that position
if the whole range were sorted with respect to \tcode{comp} and \tcode{proj},
unless \tcode{nth == last}.
Also for every iterator \tcode{i} in the range \range{first}{nth}
and every iterator \tcode{j} in the range \range{nth}{last}
it holds that:
\tcode{bool(invoke(comp, invoke(proj, *j), invoke(proj, *i)))} is \tcode{false}.

\pnum
\returns
\tcode{last} for the overload in namespace \tcode{ranges}.

\pnum
\complexity
For the overloads with no \tcode{ExecutionPolicy}, linear on average.
For the overloads with an \tcode{ExecutionPolicy}, \bigoh{N} applications of
the predicate, and \bigoh{N \log N} swaps, where $N = \tcode{last - first}$.
\end{itemdescr}

\begin{itemdecl}
template<RandomAccessRange R, class Comp = ranges::less, class Proj = identity>
  requires Sortable<iterator_t<R>, Comp, Proj>
  constexpr safe_iterator_t<R>
    ranges::nth_element(R&& r, iterator_t<R> nth, Comp comp = {}, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
\effects
Equivalent to:
\begin{codeblock}
return ranges::nth_element(ranges::begin(r), nth, ranges::end(r), comp, proj);
\end{codeblock}
\end{itemdescr}

\rSec2[alg.binary.search]{Binary search}

\pnum
All of the algorithms in this subclause are versions of binary search and
assume that the sequence being searched
is partitioned with respect to an expression
formed by binding the search key to an argument of the comparison function.
They work on non-random access iterators minimizing the number of comparisons,
which will be logarithmic for all types of iterators.
They are especially appropriate for random access iterators,
because these algorithms do a logarithmic number of steps
through the data structure.
For non-random access iterators they execute a linear number of steps.

\rSec3[lower.bound]{\tcode{lower_bound}}

\indexlibrary{\idxcode{lower_bound}}%
\begin{itemdecl}
template<class ForwardIterator, class T>
  constexpr ForwardIterator
    lower_bound(ForwardIterator first, ForwardIterator last,
                const T& value);

template<class ForwardIterator, class T, class Compare>
  constexpr ForwardIterator
    lower_bound(ForwardIterator first, ForwardIterator last,
                const T& value, Compare comp);

template<ForwardIterator I, Sentinel<I> S, class T, class Proj = identity,
         IndirectStrictWeakOrder<const T*, projected<I, Proj>> Comp = ranges::less>
  constexpr I ranges::lower_bound(I first, S last, const T& value, Comp comp = {},
                                  Proj proj = {});
template<ForwardRange R, class T, class Proj = identity,
         IndirectStrictWeakOrder<const T*, projected<iterator_t<R>, Proj>> Comp =
           ranges::less>
  constexpr safe_iterator_t<R>
    ranges::lower_bound(R&& r, const T& value, Comp comp = {}, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{comp} be \tcode{less\{\}} and
\tcode{proj} be \tcode{identity\{\}}
for overloads with no parameters by those names.

\pnum
\requires
The elements \tcode{e} of \range{first}{last}
shall be partitioned with respect to the expression
\tcode{bool(invoke(comp, invoke(proj, e), value))}.

\pnum
\returns
The furthermost iterator \tcode{i} in the range \crange{first}{last}
such that for every iterator \tcode{j} in the range \range{first}{i},
\tcode{bool(invoke(comp, invoke(proj, *j), value))} is \tcode{true}.

\pnum
\complexity
At most $\log_2(\tcode{last - first}) + \bigoh{1}$ comparisons and projections.
\end{itemdescr}

\rSec3[upper.bound]{\tcode{upper_bound}}

\indexlibrary{\idxcode{upper_bound}}%
\begin{itemdecl}
template<class ForwardIterator, class T>
  constexpr ForwardIterator
    upper_bound(ForwardIterator first, ForwardIterator last,
                const T& value);

template<class ForwardIterator, class T, class Compare>
  constexpr ForwardIterator
    upper_bound(ForwardIterator first, ForwardIterator last,
                const T& value, Compare comp);

template<ForwardIterator I, Sentinel<I> S, class T, class Proj = identity,
         IndirectStrictWeakOrder<const T*, projected<I, Proj>> Comp = ranges::less>
  constexpr I ranges::upper_bound(I first, S last, const T& value, Comp comp = {}, Proj proj = {});
template<ForwardRange R, class T, class Proj = identity,
         IndirectStrictWeakOrder<const T*, projected<iterator_t<R>, Proj>> Comp =
           ranges::less>
  constexpr safe_iterator_t<R>
    ranges::upper_bound(R&& r, const T& value, Comp comp = {}, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{comp} be \tcode{less\{\}} and
\tcode{proj} be \tcode{identity\{\}}
for overloads with no parameters by those names.

\pnum
\requires
The elements \tcode{e} of \range{first}{last}
shall be partitioned with respect to the expression
\tcode{!bool(invoke(comp, value, invoke(proj, e)))}.

\pnum
\returns
The furthermost iterator \tcode{i} in the range \crange{first}{last}
such that for every iterator \tcode{j} in the range \range{first}{i},
\tcode{!bool(invoke(comp, value, invoke(proj, *j)))} is \tcode{true}.

\pnum
\complexity
At most $\log_2(\tcode{last - first}) + \bigoh{1}$ comparisons and projections.
\end{itemdescr}

\rSec3[equal.range]{\tcode{equal_range}}

\indexlibrary{\idxcode{equal_range}}%
\begin{itemdecl}
template<class ForwardIterator, class T>
  constexpr pair<ForwardIterator, ForwardIterator>
    equal_range(ForwardIterator first,
                ForwardIterator last, const T& value);

template<class ForwardIterator, class T, class Compare>
  constexpr pair<ForwardIterator, ForwardIterator>
    equal_range(ForwardIterator first,
                ForwardIterator last, const T& value,
                Compare comp);

template<ForwardIterator I, Sentinel<I> S, class T, class Proj = identity,
         IndirectStrictWeakOrder<const T*, projected<I, Proj>> Comp = ranges::less>
  constexpr subrange<I>
    ranges::equal_range(I first, S last, const T& value, Comp comp = {}, Proj proj = {});
template<ForwardRange R, class T, class Proj = identity,
         IndirectStrictWeakOrder<const T*, projected<iterator_t<R>, Proj>> Comp =
           ranges::less>
  constexpr safe_subrange_t<R>
    ranges::equal_range(R&& r, const T& value, Comp comp = {}, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{comp} be \tcode{less\{\}} and
\tcode{proj} be \tcode{identity\{\}}
for overloads with no parameters by those names.

\pnum
\requires
The elements \tcode{e} of \range{first}{last}
shall be partitioned with respect to the expressions
\tcode{bool(invoke(comp, invoke(proj, e), value))} and
\tcode{!bool(invoke(comp, value, invoke(proj, e)))}.
Also, for all elements \tcode{e} of \tcode{[first, last)},
\tcode{bool(comp(e, value))} shall imply \tcode{!bool(comp(\brk{}value, e))}
for the overloads in namespace \tcode{std}.

\pnum
\returns
\begin{itemize}
\item
For the overloads in namespace \tcode{std}:
\begin{codeblock}
{lower_bound(first, last, value, comp),
 upper_bound(first, last, value, comp)}
\end{codeblock}
\item
For the overloads in namespace \tcode{ranges}:
\begin{codeblock}
{ranges::lower_bound(first, last, value, comp, proj),
 ranges::upper_bound(first, last, value, comp, proj)}
\end{codeblock}
\end{itemize}

\pnum
\complexity
At most
$2 * \log_2(\tcode{last - first}) + \bigoh{1}$ comparisons and projections.
\end{itemdescr}

\rSec3[binary.search]{\tcode{binary_search}}

\indexlibrary{\idxcode{binary_search}}%
\begin{itemdecl}
template<class ForwardIterator, class T>
  constexpr bool
    binary_search(ForwardIterator first, ForwardIterator last,
                  const T& value);

template<class ForwardIterator, class T, class Compare>
  constexpr bool
    binary_search(ForwardIterator first, ForwardIterator last,
                  const T& value, Compare comp);

template<ForwardIterator I, Sentinel<I> S, class T, class Proj = identity,
         IndirectStrictWeakOrder<const T*, projected<I, Proj>> Comp = ranges::less>
  constexpr bool ranges::binary_search(I first, S last, const T& value, Comp comp = {},
                                       Proj proj = {});
template<ForwardRange R, class T, class Proj = identity,
         IndirectStrictWeakOrder<const T*, projected<iterator_t<R>, Proj>> Comp =
           ranges::less>
  constexpr bool ranges::binary_search(R&& r, const T& value, Comp comp = {},
                                       Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{comp} be \tcode{less\{\}} and
\tcode{proj} be \tcode{identity\{\}}
for overloads with no parameters by those names.

\pnum
\requires
The elements \tcode{e} of \range{first}{last}
shall be partitioned with respect to the expressions
\tcode{bool(invoke(comp, invoke(proj, e), value))} and
\tcode{!bool(invoke(comp, value, invoke(proj, e)))}.
Also, for all elements \tcode{e} of \tcode{[first, last)},
\tcode{bool(comp(e, value))} shall imply \tcode{!bool(comp(\brk{}value, e))}
for the overloads in namespace \tcode{std}.

\pnum
\returns
\tcode{true} if and only if
for some iterator \tcode{i} in the range \range{first}{last},
\tcode{!bool(invoke(comp, invoke(proj, *i), value)) \&\&
!bool(invoke(comp, value, invoke(proj, *i)))}
is \tcode{true}.

\pnum
\complexity
At most $\log_2(\tcode{last - first}) + \bigoh{1}$ comparisons and projections.
\end{itemdescr}

\rSec2[alg.partitions]{Partitions}

\indexlibrary{\idxcode{is_partitioned}}%
\begin{itemdecl}
template<class InputIterator, class Predicate>
  constexpr bool is_partitioned(InputIterator first, InputIterator last, Predicate pred);
template<class ExecutionPolicy, class ForwardIterator, class Predicate>
  bool is_partitioned(ExecutionPolicy&& exec,
                      ForwardIterator first, ForwardIterator last, Predicate pred);

template<InputIterator I, Sentinel<I> S, class Proj = identity,
         IndirectUnaryPredicate<projected<I, Proj>> Pred>
  constexpr bool ranges::is_partitioned(I first, S last, Pred pred, Proj proj = {});
template<InputRange R, class Proj = identity,
         IndirectUnaryPredicate<projected<iterator_t<R>, Proj>> Pred>
  constexpr bool ranges::is_partitioned(R&& r, Pred pred, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{proj} be \tcode{identity\{\}}
for the overloads with no parameter named \tcode{proj}.

\pnum
\returns
\tcode{true} if and only if the elements \tcode{e} of \range{first}{last}
are partitioned with respect to the expression
\tcode{bool(invoke(pred, invoke(proj, e)))}.

\pnum
\complexity
Linear.
At most \tcode{last - first} applications of \tcode{pred} and \tcode{proj}.
\end{itemdescr}

\indexlibrary{\idxcode{partition}}%
\begin{itemdecl}
template<class ForwardIterator, class Predicate>
  constexpr ForwardIterator
    partition(ForwardIterator first, ForwardIterator last, Predicate pred);
template<class ExecutionPolicy, class ForwardIterator, class Predicate>
  ForwardIterator
    partition(ExecutionPolicy&& exec,
              ForwardIterator first, ForwardIterator last, Predicate pred);

template<Permutable I, Sentinel<I> S, class Proj = identity,
         IndirectUnaryPredicate<projected<I, Proj>> Pred>
  constexpr I
    ranges::partition(I first, S last, Pred pred, Proj proj = {});
template<ForwardRange R, class Proj = identity,
         IndirectUnaryPredicate<projected<iterator_t<R>, Proj>> Pred>
  requires Permutable<iterator_t<R>>
  constexpr safe_iterator_t<R>
    ranges::partition(R&& r, Pred pred, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{proj} be \tcode{identity\{\}}
for the overloads with no parameter named \tcode{proj}
and let $E(x)$ be \tcode{bool(invoke(\brk{}pred, invoke(proj, $x$)))}.

\pnum
\requires
For the overloads in namespace \tcode{std},
\tcode{ForwardIterator} shall meet
the \oldconcept{ValueSwappable} requirements\iref{swappable.requirements}.

\pnum
\effects
Places all the elements \tcode{e} in \range{first}{last}
that satisfy $E(\tcode{e})$ before all the elements that do not.

\pnum
\returns
An iterator \tcode{i} such that $E(\tcode{*j})$ is
\tcode{true} for every iterator \tcode{j} in \range{first}{i} and
\tcode{false} for every iterator \tcode{j} in \range{i}{last}.

\pnum
\complexity Let $N = \tcode{last - first}$:
\begin{itemize}
\item
  For the overload with no \tcode{ExecutionPolicy},
  exactly $N$ applications of the predicate and projection.
  At most $N / 2$ swaps if the type of \tcode{first} meets
  the \oldconcept{BidirectionalIterator} requirements
  for the overloads in namespace \tcode{std} or
  models \libconcept{BidirectionalIterator}
  for the overloads in namespace \tcode{ranges},
  and at most $N$ swaps otherwise.
\item
  For the overload with an \tcode{ExecutionPolicy},
  \bigoh{N \log N} swaps and \bigoh{N} applications of the predicate.
\end{itemize}

\end{itemdescr}

\indexlibrary{\idxcode{stable_partition}}%
\begin{itemdecl}
template<class BidirectionalIterator, class Predicate>
  BidirectionalIterator
    stable_partition(BidirectionalIterator first, BidirectionalIterator last, Predicate pred);
template<class ExecutionPolicy, class BidirectionalIterator, class Predicate>
  BidirectionalIterator
    stable_partition(ExecutionPolicy&& exec,
                     BidirectionalIterator first, BidirectionalIterator last, Predicate pred);

template<BidirectionalIterator I, Sentinel<I> S, class Proj = identity,
         IndirectUnaryPredicate<projected<I, Proj>> Pred>
  requires Permutable<I>
  I ranges::stable_partition(I first, S last, Pred pred, Proj proj = {});
template<BidirectionalRange R, class Proj = identity,
         IndirectUnaryPredicate<projected<iterator_t<R>, Proj>> Pred>
  requires Permutable<iterator_t<R>>
  safe_iterator_t<R> ranges::stable_partition(R&& r, Pred pred, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{proj} be \tcode{identity\{\}}
for the overloads with no parameter named \tcode{proj}
and let $E(x)$ be \tcode{bool(invoke(\brk{}pred, invoke(proj, $x$)))}.

\requires
For the overloads in namespace \tcode{std},
\tcode{BidirectionalIterator} shall meet
the \oldconcept{Value\-Swappable} requirements\iref{swappable.requirements} and
the type of \tcode{*first} shall meet
the \oldconcept{MoveConstructible} (\tref{cpp17.moveconstructible}) and
\oldconcept{MoveAssignable} (\tref{cpp17.moveassignable}) requirements.

\pnum
\effects
Places all the elements \tcode{e} in \range{first}{last}
that satisfy $E(\tcode{e})$ before all the elements that do not.
The relative order of the elements in both groups is preserved.

\pnum
\returns
An iterator \tcode{i}
such that for every iterator \tcode{j} in \range{first}{i},
$E(\tcode{*j})$ is \tcode{true},
and for every iterator \tcode{j} in the range \range{i}{last},
$E(\tcode{*j})$ is \tcode{false},

\pnum
\complexity
Let $N$ = \tcode{last - first}:
\begin{itemize}
\item
  For the overloads with no \tcode{ExecutionPolicy}, at most $N \log N$ swaps,
  but only \bigoh{N} swaps if there is enough extra memory.
  Exactly $N$ applications of the predicate and projection.
\item
  For the overload with an \tcode{ExecutionPolicy},
  \bigoh{N \log N} swaps and \bigoh{N} applications of the predicate.
\end{itemize}
\end{itemdescr}

\indexlibrary{\idxcode{partition_copy}}%
\begin{itemdecl}
template<class InputIterator, class OutputIterator1,
         class OutputIterator2, class Predicate>
  constexpr pair<OutputIterator1, OutputIterator2>
    partition_copy(InputIterator first, InputIterator last,
                   OutputIterator1 out_true, OutputIterator2 out_false, Predicate pred);
template<class ExecutionPolicy, class ForwardIterator, class ForwardIterator1,
         class ForwardIterator2, class Predicate>
  pair<ForwardIterator1, ForwardIterator2>
    partition_copy(ExecutionPolicy&& exec,
                   ForwardIterator first, ForwardIterator last,
                   ForwardIterator1 out_true, ForwardIterator2 out_false, Predicate pred);

template<InputIterator I, Sentinel<I> S, WeaklyIncrementable O1, WeaklyIncrementable O2,
         class Proj = identity, IndirectUnaryPredicate<projected<I, Proj>> Pred>
  requires IndirectlyCopyable<I, O1> && IndirectlyCopyable<I, O2>
  constexpr ranges::partition_copy_result<I, O1, O2>
    ranges::partition_copy(I first, S last, O1 out_true, O2 out_false, Pred pred,
                   Proj proj = {});
template<InputRange R, WeaklyIncrementable O1, WeaklyIncrementable O2,
         class Proj = identity,
         IndirectUnaryPredicate<projected<iterator_t<R>, Proj>> Pred>
  requires IndirectlyCopyable<iterator_t<R>, O1> &&
           IndirectlyCopyable<iterator_t<R>, O2>
  constexpr ranges::partition_copy_result<safe_iterator_t<R>, O1, O2>
    ranges::partition_copy(R&& r, O1 out_true, O2 out_false, Pred pred, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{proj} be \tcode{identity\{\}}
for the overloads with no parameter named \tcode{proj} and
let $E(x)$ be \tcode{bool(invoke(\brk{}pred, invoke(proj, $x$)))}.

\pnum
\requires
The input range and output ranges shall not overlap.
For the overloads in namespace \tcode{std},
the expression \tcode{*first}
shall be writable\iref{iterator.requirements.general}
to \tcode{out_true} and \tcode{out_false}.
\begin{note}
For the overload with an \tcode{ExecutionPolicy},
there may be a performance cost if \tcode{first}'s value type
does not meet the \oldconcept{CopyConstructible} requirements.
\end{note}

\pnum
\effects
For each iterator \tcode{i} in \range{first}{last},
copies \tcode{*i} to the output range beginning with \tcode{out_true}
if \tcode{$E(\tcode{*i})$} is \tcode{true}, or
to the output range beginning with \tcode{out_false} otherwise.

\pnum
\returns
Let \tcode{o1} be the end of the output range beginning at \tcode{out_true},
and \tcode{o2} the end of the output range beginning at \tcode{out_false}.
Returns
\begin{itemize}
\item \tcode{\{o1, o2\}} for the overloads in namespace \tcode{std}, or
\item \tcode{\{last, o1, o2\}} for the overloads in namespace \tcode{ranges}.
\end{itemize}

\pnum
\complexity
Exactly \tcode{last - first} applications of \tcode{pred} and \tcode{proj}.
\end{itemdescr}

\indexlibrary{\idxcode{partition_point}}%
\begin{itemdecl}
template<class ForwardIterator, class Predicate>
  constexpr ForwardIterator
    partition_point(ForwardIterator first, ForwardIterator last, Predicate pred);

template<ForwardIterator I, Sentinel<I> S, class Proj = identity,
         IndirectUnaryPredicate<projected<I, Proj>> Pred>
  constexpr I ranges::partition_point(I first, S last, Pred pred, Proj proj = {});
template<ForwardRange R, class Proj = identity,
         IndirectUnaryPredicate<projected<iterator_t<R>, Proj>> Pred>
  constexpr safe_iterator_t<R>
    ranges::partition_point(R&& r, Pred pred, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{proj} be \tcode{identity\{\}}
for the overloads with no parameter named \tcode{proj}
and let $E(x)$ be \tcode{bool(invoke(\brk{}pred, invoke(proj, $x$)))}.

\pnum
\requires
The elements \tcode{e} of \range{first}{last}
shall be partitioned with respect to $E(\tcode{e})$.

\pnum
\returns
An iterator \tcode{mid}
such that $E(\tcode{*i})$ is \tcode{true}
for all iterators \tcode{i} in \range{first}{mid}, and
\tcode{false} for all iterators \tcode{i} in \range{mid}{last}

\pnum
\complexity
\bigoh{\log(\tcode{last - first})} applications
of \tcode{pred} and \tcode{proj}.
\end{itemdescr}

\rSec2[alg.merge]{Merge}

\indexlibrary{\idxcode{merge}}%
\begin{itemdecl}
template<class InputIterator1, class InputIterator2,
         class OutputIterator>
  constexpr OutputIterator
    merge(InputIterator1 first1, InputIterator1 last1,
          InputIterator2 first2, InputIterator2 last2,
          OutputIterator result);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
         class ForwardIterator>
  ForwardIterator
    merge(ExecutionPolicy&& exec,
          ForwardIterator1 first1, ForwardIterator1 last1,
          ForwardIterator2 first2, ForwardIterator2 last2,
          ForwardIterator result);

template<class InputIterator1, class InputIterator2,
         class OutputIterator, class Compare>
  constexpr OutputIterator
    merge(InputIterator1 first1, InputIterator1 last1,
          InputIterator2 first2, InputIterator2 last2,
          OutputIterator result, Compare comp);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
         class ForwardIterator, class Compare>
  ForwardIterator
    merge(ExecutionPolicy&& exec,
          ForwardIterator1 first1, ForwardIterator1 last1,
          ForwardIterator2 first2, ForwardIterator2 last2,
          ForwardIterator result, Compare comp);

template<InputIterator I1, Sentinel<I1> S1, InputIterator I2, Sentinel<I2> S2,
         WeaklyIncrementable O, class Comp = ranges::less, class Proj1 = identity,
         class Proj2 = identity>
  requires Mergeable<I1, I2, O, Comp, Proj1, Proj2>
  constexpr ranges::merge_result<I1, I2, O>
    ranges::merge(I1 first1, S1 last1, I2 first2, S2 last2, O result,
                  Comp comp = {}, Proj1 proj1 = {}, Proj2 proj2 = {});
template<InputRange R1, InputRange R2, WeaklyIncrementable O, class Comp = ranges::less,
         class Proj1 = identity, class Proj2 = identity>
  requires Mergeable<iterator_t<R1>, iterator_t<R2>, O, Comp, Proj1, Proj2>
  constexpr ranges::merge_result<safe_iterator_t<R1>, safe_iterator_t<R2>, O>
    ranges::merge(R1&& r1, R2&& r2, O result,
                  Comp comp = {}, Proj1 proj1 = {}, Proj2 proj2 = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let $N$ be \tcode{(last1 - first1) + (last2 - first2)}.
Let \tcode{comp} be \tcode{less\{\}},
\tcode{proj1} be \tcode{identity\{\}}, and
\tcode{proj2} be \tcode{identity\{\}},
for the overloads with no parameters by those names.

\pnum
\requires
The ranges \range{first1}{last1} and \range{first2}{last2}
shall be sorted with respect to \tcode{comp} and \tcode{proj1} or \tcode{proj2},
respectively.
The resulting range shall not overlap with either of the original ranges.

\pnum
\effects
Copies all the elements of the two ranges \range{first1}{last1} and
\range{first2}{last2} into the range \range{result}{result_last},
where \tcode{result_last} is \tcode{result + $N$}.
If an element \tcode{a} precedes \tcode{b} in an input range,
\tcode{a} is copied into the output range before \tcode{b}.
If \tcode{e1} is an element of \range{first1}{last1} and
\tcode{e2} of \range{first2}{last2},
\tcode{e2} is copied into the output range before \tcode{e1} if and only if
\tcode{bool(invoke(comp, invoke(proj2, e2), invoke(proj1, e1)))}
is \tcode{true}.

\pnum
\returns
\begin{itemize}
\item
  \tcode{result_last}
  for the overloads in namespace \tcode{std}, or
\item
  \tcode{\{last1, last2, result_last\}}
  for the overloads in namespace \tcode{ranges}.
\end{itemize}

\pnum
\complexity
\begin{itemize}
\item
  For the overloads with no \tcode{ExecutionPolicy},
  at most $N - 1$ comparisons and applications of each projection.
\item
  For the overloads with an \tcode{ExecutionPolicy}, \bigoh{N} comparisons.
\end{itemize}

\pnum
\remarks
Stable\iref{algorithm.stable}.
\end{itemdescr}

\indexlibrary{\idxcode{inplace_merge}}%
\begin{itemdecl}
template<class BidirectionalIterator>
  void inplace_merge(BidirectionalIterator first,
                     BidirectionalIterator middle,
                     BidirectionalIterator last);
template<class ExecutionPolicy, class BidirectionalIterator>
  void inplace_merge(ExecutionPolicy&& exec,
                     BidirectionalIterator first,
                     BidirectionalIterator middle,
                     BidirectionalIterator last);

template<class BidirectionalIterator, class Compare>
  void inplace_merge(BidirectionalIterator first,
                     BidirectionalIterator middle,
                     BidirectionalIterator last, Compare comp);
template<class ExecutionPolicy, class BidirectionalIterator, class Compare>
  void inplace_merge(ExecutionPolicy&& exec,
                     BidirectionalIterator first,
                     BidirectionalIterator middle,
                     BidirectionalIterator last, Compare comp);

template<BidirectionalIterator I, Sentinel<I> S, class Comp = ranges::less,
         class Proj = identity>
  requires Sortable<I, Comp, Proj>
  I ranges::inplace_merge(I first, I middle, S last, Comp comp = {}, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{comp} be \tcode{less\{\}}
and \tcode{proj} be \tcode{identity\{\}}
for the overloads with no parameters by those names.

\pnum
\requires
\range{first}{middle} and \range{middle}{last} shall be valid ranges
sorted with respect to \tcode{comp} and \tcode{proj}.
For the overloads in namespace \tcode{std},
\tcode{BidirectionalIterator} shall meet
the \oldconcept{Value\-Swappable} requirements\iref{swappable.requirements} and
the type of \tcode{*first} shall meet
the \oldconcept{MoveConstructible} (\tref{cpp17.moveconstructible}) and
\oldconcept{MoveAssignable} (\tref{cpp17.moveassignable}) requirements.

\pnum
\effects
Merges two sorted consecutive ranges
\range{first}{middle} and \range{middle}{last},
putting the result of the merge into the range \range{first}{last}.
The resulting range is sorted with respect to \tcode{comp} and \tcode{proj}.

\pnum
\returns
\tcode{last} for the overload in namespace \tcode{ranges}.

\pnum
\complexity
Let $N = \tcode{last - first}$:
\begin{itemize}
\item
  For the overloads with no \tcode{ExecutionPolicy}, and
  if enough additional memory is available, exactly $N - 1$ comparisons.
\item
  Otherwise, \bigoh{N \log N} comparisons.
\end{itemize}
In either case, twice as many projections as comparisons.

\pnum
\remarks
Stable\iref{algorithm.stable}.
\end{itemdescr}

\begin{itemdecl}
template<BidirectionalRange R, class Comp = ranges::less, class Proj = identity>
  requires Sortable<iterator_t<R>, Comp, Proj>
  safe_iterator_t<R>
    ranges::inplace_merge(R&& r, iterator_t<R> middle, Comp comp = {}, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
\effects
Equivalent to:
\begin{codeblock}
return ranges::inplace_merge(ranges::begin(r), middle, ranges::end(r), comp, proj);
\end{codeblock}
\end{itemdescr}

\rSec2[alg.set.operations]{Set operations on sorted structures}

\pnum
This subclause defines all the basic set operations on sorted structures.
They also work with \tcode{multiset}s\iref{multiset}
containing multiple copies of equivalent elements.
The semantics of the set operations are generalized to \tcode{multiset}s
in a standard way by defining \tcode{set_union()}
to contain the maximum number of occurrences of every element,
\tcode{set_intersection()} to contain the minimum, and so on.

\rSec3[includes]{\tcode{includes}}

\indexlibrary{\idxcode{includes}}%
\begin{itemdecl}
template<class InputIterator1, class InputIterator2>
  constexpr bool includes(InputIterator1 first1, InputIterator1 last1,
                          InputIterator2 first2, InputIterator2 last2);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2>
  bool includes(ExecutionPolicy&& exec,
                ForwardIterator1 first1, ForwardIterator1 last1,
                ForwardIterator2 first2, ForwardIterator2 last2);

template<class InputIterator1, class InputIterator2, class Compare>
  constexpr bool includes(InputIterator1 first1, InputIterator1 last1,
                          InputIterator2 first2, InputIterator2 last2,
                          Compare comp);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2, class Compare>
  bool includes(ExecutionPolicy&& exec,
                ForwardIterator1 first1, ForwardIterator1 last1,
                ForwardIterator2 first2, ForwardIterator2 last2,
                Compare comp);

template<InputIterator I1, Sentinel<I1> S1, InputIterator I2, Sentinel<I2> S2,
         class Proj1 = identity, class Proj2 = identity,
         IndirectStrictWeakOrder<projected<I1, Proj1>,
                                 projected<I2, Proj2>> Comp = ranges::less>
  constexpr bool ranges::includes(I1 first1, S1 last1, I2 first2, S2 last2, Comp comp = {},
                                  Proj1 proj1 = {}, Proj2 proj2 = {});
template<InputRange R1, InputRange R2, class Proj1 = identity,
         class Proj2 = identity,
         IndirectStrictWeakOrder<projected<iterator_t<R1>, Proj1>,
                                 projected<iterator_t<R2>, Proj2>> Comp = ranges::less>
  constexpr bool ranges::includes(R1&& r1, R2&& r2, Comp comp = {},
                                  Proj1 proj1 = {}, Proj2 proj2 = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{comp} be \tcode{less\{\}},
\tcode{proj1} be \tcode{identity\{\}}, and
\tcode{proj2} be \tcode{identity\{\}},
for the overloads with no parameters by those names.

\pnum
\requires
The ranges \range{first1}{last1} and \range{first2}{last2} shall be sorted
with respect to \tcode{comp} and \tcode{proj1} or \tcode{proj2}, respectively.

\pnum
\returns
\tcode{true}
if and only if \range{first2}{last2} is a subsequence of \range{first1}{last1}.
\begin{note}
A sequence $S$ is a subsequence of another sequence $T$ if $S$ can be obtained
from $T$ by removing some, all, or none of $T$'s elements and keeping the
remaining elements in the same order.
\end{note}

\pnum
\complexity
At most \tcode{2 * ((last1 - first1) + (last2 - first2)) - 1}
comparisons and applications of each projection.
\end{itemdescr}

\rSec3[set.union]{\tcode{set_union}}

\indexlibrary{\idxcode{set_union}}%
\begin{itemdecl}
template<class InputIterator1, class InputIterator2,
         class OutputIterator>
  constexpr OutputIterator
    set_union(InputIterator1 first1, InputIterator1 last1,
              InputIterator2 first2, InputIterator2 last2,
              OutputIterator result);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
         class ForwardIterator>
  ForwardIterator
    set_union(ExecutionPolicy&& exec,
              ForwardIterator1 first1, ForwardIterator1 last1,
              ForwardIterator2 first2, ForwardIterator2 last2,
              ForwardIterator result);

template<class InputIterator1, class InputIterator2,
         class OutputIterator, class Compare>
  constexpr OutputIterator
    set_union(InputIterator1 first1, InputIterator1 last1,
              InputIterator2 first2, InputIterator2 last2,
              OutputIterator result, Compare comp);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
         class ForwardIterator, class Compare>
  ForwardIterator
    set_union(ExecutionPolicy&& exec,
              ForwardIterator1 first1, ForwardIterator1 last1,
              ForwardIterator2 first2, ForwardIterator2 last2,
              ForwardIterator result, Compare comp);

template<InputIterator I1, Sentinel<I1> S1, InputIterator I2, Sentinel<I2> S2,
         WeaklyIncrementable O, class Comp = ranges::less,
         class Proj1 = identity, class Proj2 = identity>
  requires Mergeable<I1, I2, O, Comp, Proj1, Proj2>
  constexpr ranges::set_union_result<I1, I2, O>
    ranges::set_union(I1 first1, S1 last1, I2 first2, S2 last2, O result, Comp comp = {},
                      Proj1 proj1 = {}, Proj2 proj2 = {});
template<InputRange R1, InputRange R2, WeaklyIncrementable O,
         class Comp = ranges::less, class Proj1 = identity, class Proj2 = identity>
  requires Mergeable<iterator_t<R1>, iterator_t<R2>, O, Comp, Proj1, Proj2>
  constexpr ranges::set_union_result<safe_iterator_t<R1>, safe_iterator_t<R2>, O>
    ranges::set_union(R1&& r1, R2&& r2, O result, Comp comp = {},
                      Proj1 proj1 = {}, Proj2 proj2 = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{comp} be \tcode{less\{\}},
and \tcode{proj1} and \tcode{proj2} be \tcode{identity\{\}}
for the overloads with no parameters by those names.

\pnum
\requires
The ranges \range{first1}{last1} and \range{first2}{last2} shall be sorted
with respect to \tcode{comp} and \tcode{proj1} or \tcode{proj2}, respectively.
The resulting range shall not overlap with either of the original ranges.

\pnum
\effects
Constructs a sorted union of the elements from the two ranges;
that is, the set of elements that are present in one or both of the ranges.

\pnum
\returns
Let \tcode{result_last} be the end of the constructed range.
Returns
\begin{itemize}
\item
  \tcode{result_last}
  for the overloads in namespace \tcode{std}, or
\item
  \tcode{\{last1, last2, result_last\}}
  for the overloads in namespace \tcode{ranges}.
\end{itemize}

\pnum
\complexity
At most \tcode{2 * ((last1 - first1) + (last2 - first2)) - 1}
comparisons and applications of each projection.

\pnum
\remarks
Stable\iref{algorithm.stable}.
If \range{first1}{last1} contains $m$ elements
that are equivalent to each other and
\range{first2}{last2} contains $n$ elements
that are equivalent to them,
then all $m$ elements from the first range
are copied to the output range, in order, and
then the final $\max(n - m, 0)$ elements from the second range
are copied to the output range, in order.
\end{itemdescr}

\rSec3[set.intersection]{\tcode{set_intersection}}

\indexlibrary{\idxcode{set_intersection}}%
\begin{itemdecl}
template<class InputIterator1, class InputIterator2,
         class OutputIterator>
  constexpr OutputIterator
    set_intersection(InputIterator1 first1, InputIterator1 last1,
                     InputIterator2 first2, InputIterator2 last2,
                     OutputIterator result);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
         class ForwardIterator>
  ForwardIterator
    set_intersection(ExecutionPolicy&& exec,
                     ForwardIterator1 first1, ForwardIterator1 last1,
                     ForwardIterator2 first2, ForwardIterator2 last2,
                     ForwardIterator result);

template<class InputIterator1, class InputIterator2,
         class OutputIterator, class Compare>
  constexpr OutputIterator
    set_intersection(InputIterator1 first1, InputIterator1 last1,
                     InputIterator2 first2, InputIterator2 last2,
                     OutputIterator result, Compare comp);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
         class ForwardIterator, class Compare>
  ForwardIterator
    set_intersection(ExecutionPolicy&& exec,
                     ForwardIterator1 first1, ForwardIterator1 last1,
                     ForwardIterator2 first2, ForwardIterator2 last2,
                     ForwardIterator result, Compare comp);

template<InputIterator I1, Sentinel<I1> S1, InputIterator I2, Sentinel<I2> S2,
         WeaklyIncrementable O, class Comp = ranges::less,
         class Proj1 = identity, class Proj2 = identity>
  requires Mergeable<I1, I2, O, Comp, Proj1, Proj2>
  constexpr ranges::set_intersection_result<I1, I2, O>
    ranges::set_intersection(I1 first1, S1 last1, I2 first2, S2 last2, O result,
                             Comp comp = {}, Proj1 proj1 = {}, Proj2 proj2 = {});
template<InputRange R1, InputRange R2, WeaklyIncrementable O,
         class Comp = ranges::less, class Proj1 = identity, class Proj2 = identity>
  requires Mergeable<iterator_t<R1>, iterator_t<R2>, O, Comp, Proj1, Proj2>
  constexpr ranges::set_intersection_result<safe_iterator_t<R1>, safe_iterator_t<R2>, O>
    ranges::set_intersection(R1&& r1, R2&& r2, O result,
                             Comp comp = {}, Proj1 proj1 = {}, Proj2 proj2 = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{comp} be \tcode{less\{\}},
and \tcode{proj1} and \tcode{proj2} be \tcode{identity\{\}}
for the overloads with no parameters by those names.

\pnum
\requires
The ranges \range{first1}{last1} and \range{first2}{last2} shall be sorted
with respect to \tcode{comp} and \tcode{proj1} or \tcode{proj2}, respectively.
The resulting range shall not overlap with either of the original ranges.

\pnum
\effects
Constructs a sorted intersection of the elements from the two ranges;
that is, the set of elements that are present in both of the ranges.

\pnum
\returns
Let \tcode{result_last} be the end of the constructed range.
Returns
\begin{itemize}
\item
  \tcode{result_last}
  for the overloads in namespace \tcode{std}, or
\item
  \tcode{\{last1, last2, result_last\}}
  for the overloads in namespace \tcode{ranges}.
\end{itemize}

\pnum
\complexity
At most \tcode{2 * ((last1 - first1) + (last2 - first2)) - 1}
comparisons and applications of each projection.

\pnum
\remarks
Stable\iref{algorithm.stable}.
If \range{first1}{last1} contains $m$ elements
that are equivalent to each other and
\range{first2}{last2} contains $n$ elements
that are equivalent to them,
the first $\min(m, n)$ elements
are copied from the first range to the output range, in order.
\end{itemdescr}

\rSec3[set.difference]{\tcode{set_difference}}

\indexlibrary{\idxcode{set_difference}}%
\begin{itemdecl}
template<class InputIterator1, class InputIterator2,
         class OutputIterator>
  constexpr OutputIterator
    set_difference(InputIterator1 first1, InputIterator1 last1,
                   InputIterator2 first2, InputIterator2 last2,
                   OutputIterator result);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
         class ForwardIterator>
  ForwardIterator
    set_difference(ExecutionPolicy&& exec,
                   ForwardIterator1 first1, ForwardIterator1 last1,
                   ForwardIterator2 first2, ForwardIterator2 last2,
                   ForwardIterator result);

template<class InputIterator1, class InputIterator2,
         class OutputIterator, class Compare>
  constexpr OutputIterator
    set_difference(InputIterator1 first1, InputIterator1 last1,
                   InputIterator2 first2, InputIterator2 last2,
                   OutputIterator result, Compare comp);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
         class ForwardIterator, class Compare>
  ForwardIterator
    set_difference(ExecutionPolicy&& exec,
                   ForwardIterator1 first1, ForwardIterator1 last1,
                   ForwardIterator2 first2, ForwardIterator2 last2,
                   ForwardIterator result, Compare comp);

template<InputIterator I1, Sentinel<I1> S1, InputIterator I2, Sentinel<I2> S2,
         WeaklyIncrementable O, class Comp = ranges::less,
         class Proj1 = identity, class Proj2 = identity>
  requires Mergeable<I1, I2, O, Comp, Proj1, Proj2>
  constexpr ranges::set_difference_result<I1, O>
    ranges::set_difference(I1 first1, S1 last1, I2 first2, S2 last2, O result,
                           Comp comp = {}, Proj1 proj1 = {}, Proj2 proj2 = {});
template<InputRange R1, InputRange R2, WeaklyIncrementable O,
         class Comp = ranges::less, class Proj1 = identity, class Proj2 = identity>
  requires Mergeable<iterator_t<R1>, iterator_t<R2>, O, Comp, Proj1, Proj2>
  constexpr ranges::set_difference_result<safe_iterator_t<R1>, O>
    ranges::set_difference(R1&& r1, R2&& r2, O result,
                           Comp comp = {}, Proj1 proj1 = {}, Proj2 proj2 = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{comp} be \tcode{less\{\}},
and \tcode{proj1} and \tcode{proj2} be \tcode{identity\{\}}
for the overloads with no parameters by those names.

\pnum
\requires
The ranges \range{first1}{last1} and \range{first2}{last2} shall be sorted
with respect to \tcode{comp} and \tcode{proj1} or \tcode{proj2}, respectively.
The resulting range shall not overlap with either of the original ranges.

\pnum
\effects
Copies the elements of the range \range{first1}{last1}
which are not present in the range \range{first2}{last2}
to the range beginning at \tcode{result}.
The elements in the constructed range are sorted.

\pnum
\returns
Let \tcode{result_last} be the end of the constructed range.
Returns
\begin{itemize}
\item
  \tcode{result_last}
  for the overloads in namespace \tcode{std}, or
\item
  \tcode{\{last1, result_last\}}
  for the overloads in namespace \tcode{ranges}.
\end{itemize}

\pnum
\complexity
At most \tcode{2 * ((last1 - first1) + (last2 - first2)) - 1}
comparisons and applications of each projection.

\pnum
\remarks
If \range{first1}{last1} contains $m$ elements
that are equivalent to each other and
\range{first2}{last2} contains $n$ elements
that are equivalent to them,
the last $\max(m - n, 0)$ elements from \range{first1}{last1}
is copied to the output range, in order.
\end{itemdescr}

\rSec3[set.symmetric.difference]{\tcode{set_symmetric_difference}}

\indexlibrary{\idxcode{set_symmetric_difference}}%
\begin{itemdecl}
template<class InputIterator1, class InputIterator2,
         class OutputIterator>
  constexpr OutputIterator
    set_symmetric_difference(InputIterator1 first1, InputIterator1 last1,
                             InputIterator2 first2, InputIterator2 last2,
                             OutputIterator result);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
         class ForwardIterator>
  ForwardIterator
    set_symmetric_difference(ExecutionPolicy&& exec,
                             ForwardIterator1 first1, ForwardIterator1 last1,
                             ForwardIterator2 first2, ForwardIterator2 last2,
                             ForwardIterator result);

template<class InputIterator1, class InputIterator2,
         class OutputIterator, class Compare>
  constexpr OutputIterator
    set_symmetric_difference(InputIterator1 first1, InputIterator1 last1,
                             InputIterator2 first2, InputIterator2 last2,
                             OutputIterator result, Compare comp);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
         class ForwardIterator, class Compare>
  ForwardIterator
    set_symmetric_difference(ExecutionPolicy&& exec,
                             ForwardIterator1 first1, ForwardIterator1 last1,
                             ForwardIterator2 first2, ForwardIterator2 last2,
                             ForwardIterator result, Compare comp);

template<InputIterator I1, Sentinel<I1> S1, InputIterator I2, Sentinel<I2> S2,
         WeaklyIncrementable O, class Comp = ranges::less,
         class Proj1 = identity, class Proj2 = identity>
  requires Mergeable<I1, I2, O, Comp, Proj1, Proj2>
  constexpr ranges::set_symmetric_difference_result<I1, I2, O>
    ranges::set_symmetric_difference(I1 first1, S1 last1, I2 first2, S2 last2, O result,
                                     Comp comp = {}, Proj1 proj1 = {},
                                     Proj2 proj2 = {});
template<InputRange R1, InputRange R2, WeaklyIncrementable O,
         class Comp = ranges::less, class Proj1 = identity, class Proj2 = identity>
  requires Mergeable<iterator_t<R1>, iterator_t<R2>, O, Comp, Proj1, Proj2>
  constexpr ranges::set_symmetric_difference_result<safe_iterator_t<R1>, safe_iterator_t<R2>, O>
    ranges::set_symmetric_difference(R1&& r1, R2&& r2, O result, Comp comp = {},
                                     Proj1 proj1 = {}, Proj2 proj2 = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{comp} be \tcode{less\{\}},
and \tcode{proj1} and \tcode{proj2} be \tcode{identity\{\}}
for the overloads with no parameters by those names.

\pnum
\requires
The resulting range shall not overlap with either of the original ranges.
The ranges \range{first1}{last1} and \range{first2}{last2} shall be sorted
with respect to \tcode{comp} and \tcode{proj1} or \tcode{proj2}, respectively.
The resulting range shall not overlap with either of the original ranges.

\pnum
\effects
Copies the elements of the range \range{first1}{last1}
that are not present in the range \range{first2}{last2},
and the elements of the range \range{first2}{last2}
that are not present in the range \range{first1}{last1}
to the range beginning at \tcode{result}.
The elements in the constructed range are sorted.

\pnum
\returns
Let \tcode{result_last} be the end of the constructed range.
Returns
\begin{itemize}
\item
  \tcode{result_last}
  for the overloads in namespace \tcode{std}, or
\item
  \tcode{\{last1, last2, result_last\}}
  for the overloads in namespace \tcode{ranges}.
\end{itemize}

\pnum
\complexity
At most \tcode{2 * ((last1 - first1) + (last2 - first2)) - 1}
comparisons and applications of each projection.

\pnum
\remarks
Stable\iref{algorithm.stable}.
If \range{first1}{last1} contains $m$ elements
that are equivalent to each other and
\range{first2}{last2} contains $n$ elements
that are equivalent to them,
then $|m - n|$ of those elements shall be copied to the output range:
the last $m - n$ of these elements from \range{first1}{last1} if $m > n$, and
the last $n - m$ of these elements from \range{first2}{last2} if $m < n$.
In either case, the elements are copied in order.
\end{itemdescr}

\rSec2[alg.heap.operations]{Heap operations}

\pnum
A random access range \range{a}{b} is a
\defnx{heap with respect to \tcode{comp} and \tcode{proj}}
{heap with respect to comp and proj@heap with respect to \tcode{comp} and \tcode{proj}}
for a comparator and projection \tcode{comp} and \tcode{proj}
if its elements are organized such that:

\begin{itemize}
\item
  With \tcode{$N$ = b - a}, for all $i$, $0 < i < N$,
  \tcode{bool(invoke(comp, invoke(proj, a[$\left \lfloor{\frac{i - 1}{2}}\right \rfloor$]), invoke(\brk{}proj, a[$i$])))}
  is \tcode{false}.
\item
  \tcode{*a} may be removed by \tcode{pop_heap}, or
  a new element added by \tcode{push_heap},
  in \bigoh{\log N} time.
\end{itemize}

\pnum
These properties make heaps useful as priority queues.

\pnum
\tcode{make_heap} converts a range into a heap and
\tcode{sort_heap} turns a heap into a sorted sequence.

\rSec3[push.heap]{\tcode{push_heap}}

\indexlibrary{\idxcode{push_heap}}%
\begin{itemdecl}
template<class RandomAccessIterator>
  constexpr void push_heap(RandomAccessIterator first, RandomAccessIterator last);

template<class RandomAccessIterator, class Compare>
  constexpr void push_heap(RandomAccessIterator first, RandomAccessIterator last,
                           Compare comp);

template<RandomAccessIterator I, Sentinel<I> S, class Comp = ranges::less,
         class Proj = identity>
  requires Sortable<I, Comp, Proj>
  constexpr I
    ranges::push_heap(I first, S last, Comp comp = {}, Proj proj = {});
template<RandomAccessRange R, class Comp = ranges::less, class Proj = identity>
  requires Sortable<iterator_t<R>, Comp, Proj>
  constexpr safe_iterator_t<R>
    ranges::push_heap(R&& r, Comp comp = {}, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{comp} be \tcode{less\{\}}
and \tcode{proj} be \tcode{identity\{\}}
for the overloads with no parameters by those names.

\pnum
\requires
The range \range{first}{last - 1}
shall be a valid heap with respect to \tcode{comp} and \tcode{proj}.
For the overloads in namespace \tcode{std},
the type of \tcode{*first} shall meet
the \oldconcept{MoveConstructible} requirements (\tref{cpp17.moveconstructible}) and
the \oldconcept{MoveAssignable} requirements (\tref{cpp17.moveassignable}).

\pnum
\effects
Places the value in the location \tcode{last - 1}
into the resulting heap \range{first}{last}.

\pnum
\returns
\tcode{last} for the overloads in namespace \tcode{ranges}.

\pnum
\complexity
At most $\log(\tcode{last - first})$ comparisons and twice as many projections.
\end{itemdescr}

\rSec3[pop.heap]{\tcode{pop_heap}}

\indexlibrary{\idxcode{pop_heap}}%
\begin{itemdecl}
template<class RandomAccessIterator>
  constexpr void pop_heap(RandomAccessIterator first, RandomAccessIterator last);

template<class RandomAccessIterator, class Compare>
  constexpr void pop_heap(RandomAccessIterator first, RandomAccessIterator last,
                          Compare comp);

template<RandomAccessIterator I, Sentinel<I> S, class Comp = ranges::less,
         class Proj = identity>
  requires Sortable<I, Comp, Proj>
  constexpr I
    ranges::pop_heap(I first, S last, Comp comp = {}, Proj proj = {});
template<RandomAccessRange R, class Comp = ranges::less, class Proj = identity>
  requires Sortable<iterator_t<R>, Comp, Proj>
  constexpr safe_iterator_t<R>
    ranges::pop_heap(R&& r, Comp comp = {}, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{comp} be \tcode{less\{\}}
and \tcode{proj} be \tcode{identity\{\}}
for the overloads with no parameters by those names.

\pnum
\requires
The range \range{first}{last}
shall be a valid non-empty heap with respect to \tcode{comp} and \tcode{proj}.
For the overloads in namespace \tcode{std},
\tcode{RandomAccessIterator} shall meet
the \oldconcept{ValueSwappable} requirements\iref{swappable.requirements} and
the type of \tcode{*first} shall meet
the \oldconcept{MoveConstructible} (\tref{cpp17.moveconstructible}) and
\oldconcept{MoveAssignable} (\tref{cpp17.moveassignable}) requirements.

\pnum
\effects
Swaps the value in the location \tcode{first}
with the value in the location
\tcode{last - 1}
and makes
\range{first}{last - 1}
into a heap with respect to \tcode{comp} and \tcode{proj}.

\pnum
\returns
\tcode{last} for the overloads in namespace \tcode{ranges}.

\pnum
\complexity
At most $2 \log(\tcode{last - first})$ comparisons and
twice as many projections.
\end{itemdescr}

\rSec3[make.heap]{\tcode{make_heap}}

\indexlibrary{\idxcode{make_heap}}%
\begin{itemdecl}
template<class RandomAccessIterator>
  constexpr void make_heap(RandomAccessIterator first, RandomAccessIterator last);

template<class RandomAccessIterator, class Compare>
  constexpr void make_heap(RandomAccessIterator first, RandomAccessIterator last,
                           Compare comp);

template<RandomAccessIterator I, Sentinel<I> S, class Comp = ranges::less,
         class Proj = identity>
  requires Sortable<I, Comp, Proj>
  constexpr I
    ranges::make_heap(I first, S last, Comp comp = {}, Proj proj = {});
template<RandomAccessRange R, class Comp = ranges::less, class Proj = identity>
  requires Sortable<iterator_t<R>, Comp, Proj>
  constexpr safe_iterator_t<R>
    ranges::make_heap(R&& r, Comp comp = {}, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{comp} be \tcode{less\{\}}
and \tcode{proj} be \tcode{identity\{\}}
for the overloads with no parameters by those names.

\pnum
\requires
For the overloads in namespace \tcode{std},
the type of \tcode{*first} shall meet
the \oldconcept{Move\-Constructible} (\tref{cpp17.moveconstructible}) and
\oldconcept{MoveAssignable} (\tref{cpp17.moveassignable}) requirements.

\pnum
\effects
Constructs a heap with respect to \tcode{comp} and \tcode{proj}
out of the range \range{first}{last}.

\pnum
\returns
\tcode{last} for the overloads in namespace \tcode{ranges}.

\pnum
\complexity
At most $3(\tcode{last - first})$ comparisons and twice as many projections.
\end{itemdescr}

\rSec3[sort.heap]{\tcode{sort_heap}}

\indexlibrary{\idxcode{sort_heap}}%
\begin{itemdecl}
template<class RandomAccessIterator>
  constexpr void sort_heap(RandomAccessIterator first, RandomAccessIterator last);

template<class RandomAccessIterator, class Compare>
  constexpr void sort_heap(RandomAccessIterator first, RandomAccessIterator last,
                           Compare comp);

template<RandomAccessIterator I, Sentinel<I> S, class Comp = ranges::less,
         class Proj = identity>
  requires Sortable<I, Comp, Proj>
  constexpr I
    ranges::sort_heap(I first, S last, Comp comp = {}, Proj proj = {});
template<RandomAccessRange R, class Comp = ranges::less, class Proj = identity>
  requires Sortable<iterator_t<R>, Comp, Proj>
  constexpr safe_iterator_t<R>
    ranges::sort_heap(R&& r, Comp comp = {}, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{comp} be \tcode{less\{\}}
and \tcode{proj} be \tcode{identity\{\}}
for the overloads with no parameters by those names.

\pnum
\requires
The range \range{first}{last} shall be
a valid heap with respect to \tcode{comp} and \tcode{proj}.
For the overloads in namespace \tcode{std},
\tcode{RandomAccessIterator} shall meet
the \oldconcept{ValueSwappable} requirements\iref{swappable.requirements} and
the type of \tcode{*first} shall meet
the \oldconcept{MoveConst\-ruct\-ible} (\tref{cpp17.moveconstructible}) and
\oldconcept{MoveAssignable} (\tref{cpp17.moveassignable}) requirements.

\pnum
\effects
Sorts elements in the heap \range{first}{last}
with respect to \tcode{comp} and \tcode{proj}.

\pnum
\returns
\tcode{last} for the overloads in namespace \tcode{ranges}.

\pnum
\complexity
At most $2N \log N$ comparisons, where $N = \tcode{last - first}$, and
twice as many projections.
\end{itemdescr}

\rSec3[is.heap]{\tcode{is_heap}}

\indexlibrary{\idxcode{is_heap}}%
\begin{itemdecl}
template<class RandomAccessIterator>
  constexpr bool is_heap(RandomAccessIterator first, RandomAccessIterator last);
\end{itemdecl}

\begin{itemdescr}
\pnum
\effects
Equivalent to: \tcode{return is_heap_until(first, last) == last;}
\end{itemdescr}

\indexlibrary{\idxcode{is_heap}}%
\begin{itemdecl}
template<class ExecutionPolicy, class RandomAccessIterator>
  bool is_heap(ExecutionPolicy&& exec,
               RandomAccessIterator first, RandomAccessIterator last);
\end{itemdecl}

\begin{itemdescr}
\pnum
\effects
Equivalent to:
\begin{codeblock}
return is_heap_until(std::forward<ExecutionPolicy>(exec), first, last) == last;
\end{codeblock}
\end{itemdescr}

\indexlibrary{\idxcode{is_heap}}%
\begin{itemdecl}
template<class RandomAccessIterator, class Compare>
  constexpr bool is_heap(RandomAccessIterator first, RandomAccessIterator last,
                         Compare comp);
\end{itemdecl}

\begin{itemdescr}
\pnum
\effects
Equivalent to: \tcode{return is_heap_until(first, last, comp) == last;}
\end{itemdescr}

\indexlibrary{\idxcode{is_heap}}%
\begin{itemdecl}
template<class ExecutionPolicy, class RandomAccessIterator, class Compare>
  bool is_heap(ExecutionPolicy&& exec,
               RandomAccessIterator first, RandomAccessIterator last,
               Compare comp);
\end{itemdecl}

\begin{itemdescr}
\pnum
\effects
Equivalent to:
\begin{codeblock}
return is_heap_until(std::forward<ExecutionPolicy>(exec), first, last, comp) == last;
\end{codeblock}
\end{itemdescr}

\indexlibrary{\idxcode{is_heap}}%
\begin{itemdecl}
template<RandomAccessIterator I, Sentinel<I> S, class Proj = identity,
         IndirectStrictWeakOrder<projected<I, Proj>> Comp = ranges::less>
  constexpr bool ranges::is_heap(I first, S last, Comp comp = {}, Proj proj = {});
template<RandomAccessRange R, class Proj = identity,
         IndirectStrictWeakOrder<projected<iterator_t<R>, Proj>> Comp = ranges::less>
  constexpr bool ranges::is_heap(R&& r, Comp comp = {}, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
\effects
Equivalent to:
\tcode{return ranges::is_heap_until(first, last, comp, proj) == last;}
\end{itemdescr}

\indexlibrary{\idxcode{is_heap_until}}%
\begin{itemdecl}
template<class RandomAccessIterator>
  constexpr RandomAccessIterator
    is_heap_until(RandomAccessIterator first, RandomAccessIterator last);
template<class ExecutionPolicy, class RandomAccessIterator>
  RandomAccessIterator
    is_heap_until(ExecutionPolicy&& exec,
                  RandomAccessIterator first, RandomAccessIterator last);

template<class RandomAccessIterator, class Compare>
  constexpr RandomAccessIterator
    is_heap_until(RandomAccessIterator first, RandomAccessIterator last,
                  Compare comp);
template<class ExecutionPolicy, class RandomAccessIterator, class Compare>
  RandomAccessIterator
    is_heap_until(ExecutionPolicy&& exec,
                  RandomAccessIterator first, RandomAccessIterator last,
                  Compare comp);

template<RandomAccessIterator I, Sentinel<I> S, class Proj = identity,
         IndirectStrictWeakOrder<projected<I, Proj>> Comp = ranges::less>
  constexpr I ranges::is_heap_until(I first, S last, Comp comp = {}, Proj proj = {});
template<RandomAccessRange R, class Proj = identity,
         IndirectStrictWeakOrder<projected<iterator_t<R>, Proj>> Comp = ranges::less>
  constexpr safe_iterator_t<R>
    ranges::is_heap_until(R&& r, Comp comp = {}, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{comp} be \tcode{less\{\}}
and \tcode{proj} be \tcode{identity\{\}}
for the overloads with no parameters by those names.

\pnum
\returns
The last iterator \tcode{i} in \crange{first}{last}
for which the range \range{first}{i}
is a heap with respect to \tcode{comp} and \tcode{proj}.

\pnum
\complexity
Linear.
\end{itemdescr}


\rSec2[alg.min.max]{Minimum and maximum}

\indexlibrary{\idxcode{min}}%
\begin{itemdecl}
template<class T>
  constexpr const T& min(const T& a, const T& b);
template<class T, class Compare>
  constexpr const T& min(const T& a, const T& b, Compare comp);

template<class T, class Proj = identity,
         IndirectStrictWeakOrder<projected<const T*, Proj>> Comp = ranges::less>
  constexpr const T& ranges::min(const T& a, const T& b, Comp comp = {}, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
For the first form, type \tcode{T} shall be
\oldconcept{LessThanComparable} (\tref{cpp17.lessthancomparable}).

\pnum
\returns
The smaller value.

\pnum
\remarks
Returns the first argument when the arguments are equivalent.
An invocation may explicitly specify
an argument for the template parameter \tcode{T}
of the overloads in namespace \tcode{std}.

\pnum
\complexity
Exactly one comparison and two applications of the projection, if any.
\end{itemdescr}

\indexlibrary{\idxcode{min}}%
\begin{itemdecl}
template<class T>
  constexpr T min(initializer_list<T> r);
template<class T, class Compare>
  constexpr T min(initializer_list<T> r, Compare comp);

template<Copyable T, class Proj = identity,
         IndirectStrictWeakOrder<projected<const T*, Proj>> Comp = ranges::less>
  constexpr T ranges::min(initializer_list<T> r, Comp comp = {}, Proj proj = {});
template<InputRange R, class Proj = identity,
         IndirectStrictWeakOrder<projected<iterator_t<R>, Proj>> Comp = ranges::less>
  requires IndirectlyCopyableStorable<iterator_t<R>, iter_value_t<iterator_t<R>>*>
  constexpr iter_value_t<iterator_t<R>>
    ranges::min(R&& r, Comp comp = {}, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
\tcode{ranges::distance(r) > 0}.
For the overloads in namespace \tcode{std},
\tcode{T} shall be \oldconcept{Copy\-Constructible}.
For the first form, type \tcode{T} shall be \oldconcept{LessThanComparable}.

\pnum
\returns
The smallest value in the input range.

\pnum
\remarks
Returns a copy of the leftmost element
when several elements are equivalent to the smallest.
An invocation may explicitly specify
an argument for the template parameter \tcode{T}
of the overloads in namespace \tcode{std}.

\pnum
\complexity
Exactly \tcode{ranges::distance(r) - 1} comparisons
and twice as many applications of the projection, if any.
\end{itemdescr}

\indexlibrary{\idxcode{max}}%
\begin{itemdecl}
template<class T>
  constexpr const T& max(const T& a, const T& b);
template<class T, class Compare>
  constexpr const T& max(const T& a, const T& b, Compare comp);

template<class T, class Proj = identity,
         IndirectStrictWeakOrder<projected<const T*, Proj>> Comp = ranges::less>
  constexpr const T& ranges::max(const T& a, const T& b, Comp comp = {}, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
For the first form, type \tcode{T} shall be
\oldconcept{LessThanComparable} (\tref{cpp17.lessthancomparable}).

\pnum
\returns
The larger value.

\pnum
\remarks
Returns the first argument when the arguments are equivalent.
An invocation may explicitly specify
an argument for the template parameter \tcode{T}
of the overloads in namespace \tcode{std}.

\pnum
\complexity
Exactly one comparison and two applications of the projection, if any.
\end{itemdescr}

\indexlibrary{\idxcode{max}}%
\begin{itemdecl}
template<class T>
  constexpr T max(initializer_list<T> r);
template<class T, class Compare>
  constexpr T max(initializer_list<T> r, Compare comp);

template<Copyable T, class Proj = identity,
         IndirectStrictWeakOrder<projected<const T*, Proj>> Comp = ranges::less>
  constexpr T ranges::max(initializer_list<T> r, Comp comp = {}, Proj proj = {});
template<InputRange R, class Proj = identity,
         IndirectStrictWeakOrder<projected<iterator_t<R>, Proj>> Comp = ranges::less>
  requires IndirectlyCopyableStorable<iterator_t<R>, iter_value_t<iterator_t<R>>*>
  constexpr iter_value_t<iterator_t<R>>
    ranges::max(R&& r, Comp comp = {}, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
\tcode{ranges::distance(r) > 0}.
For the overloads in namespace \tcode{std},
\tcode{T} shall be \oldconcept{Copy\-Constructible}.
For the first form, type \tcode{T} shall be \oldconcept{LessThanComparable}.

\pnum
\returns
The largest value in the input range.

\pnum
\remarks
Returns a copy of the leftmost element
when several elements are equivalent to the largest.
An invocation may explicitly specify
an argument for the template parameter \tcode{T}
of the overloads in namespace \tcode{std}.

\pnum
\complexity
Exactly \tcode{ranges::distance(r) - 1} comparisons
and twice as many applications of the projection, if any.
\end{itemdescr}

\indexlibrary{\idxcode{minmax}}%
\begin{itemdecl}
template<class T>
  constexpr pair<const T&, const T&> minmax(const T& a, const T& b);
template<class T, class Compare>
  constexpr pair<const T&, const T&> minmax(const T& a, const T& b, Compare comp);

template<class T, class Proj = identity,
         IndirectStrictWeakOrder<projected<const T*, Proj>> Comp = ranges::less>
  constexpr ranges::minmax_result<const T&>
    ranges::minmax(const T& a, const T& b, Comp comp = {}, Proj proj = {});
\end{itemdecl}


\begin{itemdescr}
\pnum
\requires
For the first form, type \tcode{T} shall be
\oldconcept{LessThanComparable} (\tref{cpp17.lessthancomparable}).

\pnum
\returns
\tcode{\{b, a\}} if \tcode{b} is smaller than \tcode{a}, and
\tcode{\{a, b\}} otherwise.

\pnum
\remarks
An invocation may explicitly specify
an argument for the template parameter \tcode{T}
of the overloads in namespace \tcode{std}.

\pnum
\complexity
Exactly one comparison and two applications of the projection, if any.
\end{itemdescr}

\indexlibrary{\idxcode{minmax}}%
\begin{itemdecl}
template<class T>
  constexpr pair<T, T> minmax(initializer_list<T> t);
template<class T, class Compare>
  constexpr pair<T, T> minmax(initializer_list<T> t, Compare comp);

template<Copyable T, class Proj = identity,
         IndirectStrictWeakOrder<projected<const T*, Proj>> Comp = ranges::less>
  constexpr ranges::minmax_result<T>
    ranges::minmax(initializer_list<T> r, Comp comp = {}, Proj proj = {});
template<InputRange R, class Proj = identity,
         IndirectStrictWeakOrder<projected<iterator_t<R>, Proj>> Comp = ranges::less>
  requires IndirectlyCopyableStorable<iterator_t<R>, iter_value_t<iterator_t<R>>*>
  constexpr ranges::minmax_result<iter_value_t<iterator_t<R>>>
    ranges::minmax(R&& r, Comp comp = {}, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
\tcode{ranges::distance(r) > 0}.
For the overloads in namespace \tcode{std},
\tcode{T} shall be \oldconcept{Copy\-Constructible}.
For the first form, type \tcode{T} shall be \oldconcept{LessThanComparable}.

\pnum
\returns
Let \tcode{X} be the return type.
Returns \tcode{X{x, y}},
where \tcode{x} is a copy of the leftmost element with the smallest and
\tcode{y} a copy of the rightmost element with the largest value
in the input range.

\pnum
\remarks
An invocation may explicitly specify
an argument for the template parameter \tcode{T}
of the overloads in namespace \tcode{std}.

\pnum
\complexity
At most $(3/2)\tcode{ranges::distance(r)}$ applications
of the corresponding predicate
and twice as many applications of the projection, if any.
\end{itemdescr}

\indexlibrary{\idxcode{min_element}}%
\begin{itemdecl}
template<class ForwardIterator>
  constexpr ForwardIterator min_element(ForwardIterator first, ForwardIterator last);

template<class ExecutionPolicy, class ForwardIterator>
  ForwardIterator min_element(ExecutionPolicy&& exec,
                              ForwardIterator first, ForwardIterator last);

template<class ForwardIterator, class Compare>
  constexpr ForwardIterator min_element(ForwardIterator first, ForwardIterator last,
                                        Compare comp);
template<class ExecutionPolicy, class ForwardIterator, class Compare>
  ForwardIterator min_element(ExecutionPolicy&& exec,
                              ForwardIterator first, ForwardIterator last,
                              Compare comp);

template<ForwardIterator I, Sentinel<I> S, class Proj = identity,
         IndirectStrictWeakOrder<projected<I, Proj>> Comp = ranges::less>
  constexpr I ranges::min_element(I first, S last, Comp comp = {}, Proj proj = {});
template<ForwardRange R, class Proj = identity,
         IndirectStrictWeakOrder<projected<iterator_t<R>, Proj>> Comp = ranges::less>
  constexpr safe_iterator_t<R>
    ranges::min_element(R&& r, Comp comp = {}, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{comp} be \tcode{less\{\}}
and \tcode{proj} be \tcode{identity\{\}}
for the overloads with no parameters by those names.

\pnum
\returns
The first iterator \tcode{i} in the range \range{first}{last}
such that for every iterator \tcode{j} in the range \range{first}{last},
\begin{codeblock}
bool(invoke(comp, invoke(proj, *j), invoke(proj, *i)))
\end{codeblock}
is \tcode{false}.
Returns \tcode{last} if \tcode{first == last}.

\pnum
\complexity
Exactly $\max(\tcode{last - first - 1}, 0)$ comparisons and
twice as many projections.
\end{itemdescr}

\indexlibrary{\idxcode{max_element}}%
\begin{itemdecl}
template<class ForwardIterator>
  constexpr ForwardIterator max_element(ForwardIterator first, ForwardIterator last);
template<class ExecutionPolicy, class ForwardIterator>
  ForwardIterator max_element(ExecutionPolicy&& exec,
                              ForwardIterator first, ForwardIterator last);

template<class ForwardIterator, class Compare>
  constexpr ForwardIterator max_element(ForwardIterator first, ForwardIterator last,
                                        Compare comp);
template<class ExecutionPolicy, class ForwardIterator, class Compare>
  ForwardIterator max_element(ExecutionPolicy&& exec,
                              ForwardIterator first, ForwardIterator last,
                              Compare comp);

template<ForwardIterator I, Sentinel<I> S, class Proj = identity,
         IndirectStrictWeakOrder<projected<I, Proj>> Comp = ranges::less>
  constexpr I ranges::max_element(I first, S last, Comp comp = {}, Proj proj = {});
template<ForwardRange R, class Proj = identity,
         IndirectStrictWeakOrder<projected<iterator_t<R>, Proj>> Comp = ranges::less>
  constexpr safe_iterator_t<R>
    ranges::max_element(R&& r, Comp comp = {}, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{comp} be \tcode{less\{\}}
and \tcode{proj} be \tcode{identity\{\}}
for the overloads with no parameters by those names.

\pnum
\returns
The first iterator \tcode{i} in the range \range{first}{last}
such that for every iterator \tcode{j} in the range \range{first}{last},
\begin{codeblock}
bool(invoke(comp, invoke(proj, *i), invoke(proj, *j)))
\end{codeblock}
is \tcode{false}.
Returns \tcode{last} if \tcode{first == last}.

\pnum
\complexity
Exactly $\max(\tcode{last - first - 1}, 0)$ comparisons and
twice as many projections.
\end{itemdescr}

\indexlibrary{\idxcode{minmax_element}}%
\begin{itemdecl}
template<class ForwardIterator>
  constexpr pair<ForwardIterator, ForwardIterator>
    minmax_element(ForwardIterator first, ForwardIterator last);
template<class ExecutionPolicy, class ForwardIterator>
  pair<ForwardIterator, ForwardIterator>
    minmax_element(ExecutionPolicy&& exec,
                   ForwardIterator first, ForwardIterator last);

template<class ForwardIterator, class Compare>
  constexpr pair<ForwardIterator, ForwardIterator>
    minmax_element(ForwardIterator first, ForwardIterator last, Compare comp);
template<class ExecutionPolicy, class ForwardIterator, class Compare>
  pair<ForwardIterator, ForwardIterator>
    minmax_element(ExecutionPolicy&& exec,
                   ForwardIterator first, ForwardIterator last, Compare comp);

template<ForwardIterator I, Sentinel<I> S, class Proj = identity,
         IndirectStrictWeakOrder<projected<I, Proj>> Comp = ranges::less>
  constexpr ranges::minmax_result<I>
    ranges::minmax_element(I first, S last, Comp comp = {}, Proj proj = {});
template<ForwardRange R, class Proj = identity,
         IndirectStrictWeakOrder<projected<iterator_t<R>, Proj>> Comp = ranges::less>
  constexpr ranges::minmax_result<safe_iterator_t<R>>
    ranges::minmax_element(R&& r, Comp comp = {}, Proj proj = {});
\end{itemdecl}


\begin{itemdescr}
\pnum
\returns
\tcode{\{first, first\}} if \range{first}{last} is empty, otherwise
\tcode{\{m, M\}}, where \tcode{m} is
the first iterator in \range{first}{last} such that no iterator in the range refers
to a smaller element, and where \tcode{M} is the last iterator\footnote{This behavior
intentionally differs from \tcode{max_element()}.}
in \range{first}{last} such that no iterator in the range refers to a larger element.

\pnum
\complexity
Let $N$ be \tcode{last - first}.
At most $\max(\bigl\lfloor{\frac{3}{2}} (N-1)\bigr\rfloor, 0)$ comparisons and
twice as many applications of the projection, if any.
\end{itemdescr}

\rSec2[alg.clamp]{Bounded value}

\indexlibrary{\idxcode{clamp}}%
\begin{itemdecl}
template<class T>
  constexpr const T& clamp(const T& v, const T& lo, const T& hi);
template<class T, class Compare>
  constexpr const T& clamp(const T& v, const T& lo, const T& hi, Compare comp);
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
The value of \tcode{lo} shall be no greater than \tcode{hi}.
For the first form, type \tcode{T}
shall be \oldconcept{LessThan\-Comparable} (\tref{cpp17.lessthancomparable}).

\pnum
\returns
\tcode{lo} if \tcode{v} is less than \tcode{lo},
\tcode{hi} if \tcode{hi} is less than \tcode{v},
otherwise \tcode{v}.

\pnum
\begin{note}
If NaN is avoided, \tcode{T} can be a floating-point type.
\end{note}

\pnum
\complexity
At most two comparisons.
\end{itemdescr}

\rSec2[alg.lex.comparison]{Lexicographical comparison}

\indexlibrary{\idxcode{lexicographical_compare}}%
\begin{itemdecl}
template<class InputIterator1, class InputIterator2>
  constexpr bool
    lexicographical_compare(InputIterator1 first1, InputIterator1 last1,
                            InputIterator2 first2, InputIterator2 last2);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2>
  bool
    lexicographical_compare(ExecutionPolicy&& exec,
                            ForwardIterator1 first1, ForwardIterator1 last1,
                            ForwardIterator2 first2, ForwardIterator2 last2);

template<class InputIterator1, class InputIterator2, class Compare>
  constexpr bool
    lexicographical_compare(InputIterator1 first1, InputIterator1 last1,
                            InputIterator2 first2, InputIterator2 last2,
                            Compare comp);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
         class Compare>
  bool
    lexicographical_compare(ExecutionPolicy&& exec,
                            ForwardIterator1 first1, ForwardIterator1 last1,
                            ForwardIterator2 first2, ForwardIterator2 last2,
                            Compare comp);

template<InputIterator I1, Sentinel<I1> S1, InputIterator I2, Sentinel<I2> S2,
         class Proj1 = identity, class Proj2 = identity,
         IndirectStrictWeakOrder<projected<I1, Proj1>,
                                 projected<I2, Proj2>> Comp = ranges::less>
  constexpr bool
    ranges::lexicographical_compare(I1 first1, S1 last1, I2 first2, S2 last2,
                                    Comp comp = {}, Proj1 proj1 = {}, Proj2 proj2 = {});
template<InputRange R1, InputRange R2, class Proj1 = identity,
         class Proj2 = identity,
         IndirectStrictWeakOrder<projected<iterator_t<R1>, Proj1>,
                                 projected<iterator_t<R2>, Proj2>> Comp = ranges::less>
  constexpr bool
    ranges::lexicographical_compare(R1&& r1, R2&& r2, Comp comp = {},
                                    Proj1 proj1 = {}, Proj2 proj2 = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
\returns
\tcode{true} if and only if
the sequence of elements defined by the range \range{first1}{last1}
is lexicographically less than
the sequence of elements defined by the range \range{first2}{last2}.

\pnum
\complexity
At most $2 \min(\tcode{last1 - first1}, \ \tcode{last2 - first2})$ applications
of the corresponding comparison and each projection, if any.

\pnum
\remarks
If two sequences have the same number of elements and
their corresponding elements (if any) are equivalent,
then neither sequence is lexicographically less than the other.
If one sequence is a proper prefix of the other,
then the shorter sequence is lexicographically less than the longer sequence.
Otherwise, the lexicographical comparison of the sequences yields
the same result as the comparison
of the first corresponding pair of elements that are not equivalent.

\pnum
\begin{example}
\tcode{ranges::lexicographical_compare(I1, S1, I2, S2, Comp, Proj1, Proj2)}
could be implemented as:
\begin{codeblock}
for ( ; first1 != last1 && first2 != last2 ; ++first1, (void) ++first2) {
  if (invoke(comp, invoke(proj1, *first1), invoke(proj2, *first2))) return true;
  if (invoke(comp, invoke(proj2, *first2), invoke(proj1, *first1))) return false;
}
return first1 == last1 && first2 != last2;
\end{codeblock}
\end{example}

\pnum
\begin{note}
An empty sequence is lexicographically less than any non-empty sequence,
but not less than any empty sequence.
\end{note}
\end{itemdescr}

\rSec2[alg.3way]{Three-way comparison algorithms}

\indexlibrary{\idxcode{compare_3way}}%
\begin{itemdecl}
template<class T, class U> constexpr auto compare_3way(const T& a, const U& b);
\end{itemdecl}

\begin{itemdescr}
\pnum
\effects
Compares two values and produces a result
of the strongest applicable comparison category type:
\begin{itemize}
\item
  Returns \tcode{a <=> b} if that expression is well-formed.
\item
  Otherwise, if the expressions \tcode{a == b} and \tcode{a < b}
  are each well-formed and convertible to \tcode{bool},
  returns \tcode{strong_ordering::equal}
  when \tcode{a == b} is \tcode{true},
  otherwise returns \tcode{strong_ordering::less}
  when \tcode{a < b} is \tcode{true},
  and otherwise returns \tcode{strong_ordering::greater}.
\item
  Otherwise, if the expression \tcode{a == b}
  is well-formed and convertible to \tcode{bool},
  returns \tcode{strong_equality::equal} when \tcode{a == b} is \tcode{true},
  and otherwise returns \tcode{strong_equality::nonequal}.
\item
  Otherwise, the function is defined as deleted.
\end{itemize}
\end{itemdescr}

\indexlibrary{\idxcode{lexicographical_compare_3way}}%
\begin{itemdecl}
template<class InputIterator1, class InputIterator2, class Cmp>
  constexpr auto
    lexicographical_compare_3way(InputIterator1 b1, InputIterator1 e1,
                                 InputIterator2 b2, InputIterator2 e2,
                                 Cmp comp)
      -> common_comparison_category_t<decltype(comp(*b1, *b2)), strong_ordering>;
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
\tcode{Cmp} shall be a function object type
whose return type is a comparison category type.

\pnum
\effects
Lexicographically compares two ranges and
produces a result of the strongest applicable comparison category type.
Equivalent to:
\begin{codeblock}
for ( ; b1 != e1 && b2 != e2; void(++b1), void(++b2) )
  if (auto cmp = comp(*b1,*b2); cmp != 0)
    return cmp;
return b1 != e1 ? strong_ordering::greater :
       b2 != e2 ? strong_ordering::less :
                  strong_ordering::equal;
\end{codeblock}
\end{itemdescr}

\indexlibrary{\idxcode{lexicographical_compare_3way}}%
\begin{itemdecl}
template<class InputIterator1, class InputIterator2>
  constexpr auto
    lexicographical_compare_3way(InputIterator1 b1, InputIterator1 e1,
                                 InputIterator2 b2, InputIterator2 e2);
\end{itemdecl}

\begin{itemdescr}
\pnum
\effects
Equivalent to:
\begin{codeblock}
return lexicographical_compare_3way(b1, e1, b2, e2,
                                    [](const auto& t, const auto& u) {
                                      return compare_3way(t, u);
                                    });
\end{codeblock}
\end{itemdescr}

\rSec2[alg.permutation.generators]{Permutation generators}

\indexlibrary{\idxcode{next_permutation}}%
\begin{itemdecl}
template<class BidirectionalIterator>
  constexpr bool next_permutation(BidirectionalIterator first,
                                  BidirectionalIterator last);

template<class BidirectionalIterator, class Compare>
  constexpr bool next_permutation(BidirectionalIterator first,
                                  BidirectionalIterator last, Compare comp);

template<BidirectionalIterator I, Sentinel<I> S, class Comp = ranges::less,
         class Proj = identity>
  requires Sortable<I, Comp, Proj>
  constexpr bool
    ranges::next_permutation(I first, S last, Comp comp = {}, Proj proj = {});
template<BidirectionalRange R, class Comp = ranges::less,
         class Proj = identity>
  requires Sortable<iterator_t<R>, Comp, Proj>
  constexpr bool
    ranges::next_permutation(R&& r, Comp comp = {}, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{comp} be \tcode{less\{\}}
and \tcode{proj} be \tcode{identity\{\}}
for overloads with no parameters by those names.

\pnum
\requires
For the overloads in namespace \tcode{std},
\tcode{BidirectionalIterator} shall meet
the \oldconcept{Value\-Swappable} requirements\iref{swappable.requirements}.

\pnum
\effects
Takes a sequence defined by the range \range{first}{last}
and transforms it into the next permutation.
The next permutation is found by assuming that the set of all permutations
is lexicographically sorted with respect to \tcode{comp} and \tcode{proj}.
If no such permutation exists,
transforms the sequence into the first permutation;
that is, the ascendingly-sorted one.

\pnum
\returns
\tcode{true} if and only if a next permutation was found.

\pnum
\complexity
At most \tcode{(last - first) / 2} swaps.
\end{itemdescr}

\indexlibrary{\idxcode{prev_permutation}}%
\begin{itemdecl}
template<class BidirectionalIterator>
  constexpr bool prev_permutation(BidirectionalIterator first,
                                  BidirectionalIterator last);

template<class BidirectionalIterator, class Compare>
  constexpr bool prev_permutation(BidirectionalIterator first,
                                  BidirectionalIterator last, Compare comp);

template<BidirectionalIterator I, Sentinel<I> S, class Comp = ranges::less,
         class Proj = identity>
  requires Sortable<I, Comp, Proj>
  constexpr bool
    ranges::prev_permutation(I first, S last, Comp comp = {}, Proj proj = {});
template<BidirectionalRange R, class Comp = ranges::less,
         class Proj = identity>
  requires Sortable<iterator_t<R>, Comp, Proj>
  constexpr bool
    ranges::prev_permutation(R&& r, Comp comp = {}, Proj proj = {});
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{comp} be \tcode{less\{\}}
and \tcode{proj} be \tcode{identity\{\}}
for overloads with no parameters by those names.

\pnum
\requires
For the overloads in namespace \tcode{std},
\tcode{BidirectionalIterator} shall meet
the \oldconcept{Value\-Swappable} requirements\iref{swappable.requirements}.

\pnum
\effects
Takes a sequence defined by the range \range{first}{last}
and transforms it into the previous permutation.
The previous permutation is found by assuming that the set of all permutations
is lexicographically sorted with respect to \tcode{comp} and \tcode{proj}.
If no such permutation exists,
transforms the sequence into the last permutation;
that is, the descendingly-sorted one.

\pnum
\returns
\tcode{true} if and only if a previous permutation was found.

\pnum
\complexity
At most \tcode{(last - first) / 2} swaps.
\end{itemdescr}

\rSec1[numeric.ops.overview]{Header \tcode{<numeric>} synopsis}

\indexhdr{numeric}%
\begin{codeblock}
namespace std {
  // \ref{accumulate}, accumulate
  template<class InputIterator, class T>
    T accumulate(InputIterator first, InputIterator last, T init);
  template<class InputIterator, class T, class BinaryOperation>
    T accumulate(InputIterator first, InputIterator last, T init, BinaryOperation binary_op);

  // \ref{reduce}, reduce
  template<class InputIterator>
    typename iterator_traits<InputIterator>::value_type
      reduce(InputIterator first, InputIterator last);
  template<class InputIterator, class T>
    T reduce(InputIterator first, InputIterator last, T init);
  template<class InputIterator, class T, class BinaryOperation>
    T reduce(InputIterator first, InputIterator last, T init, BinaryOperation binary_op);
  template<class ExecutionPolicy, class ForwardIterator>
    typename iterator_traits<ForwardIterator>::value_type
      reduce(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
             ForwardIterator first, ForwardIterator last);
  template<class ExecutionPolicy, class ForwardIterator, class T>
    T reduce(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
             ForwardIterator first, ForwardIterator last, T init);
  template<class ExecutionPolicy, class ForwardIterator, class T, class BinaryOperation>
    T reduce(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
             ForwardIterator first, ForwardIterator last, T init, BinaryOperation binary_op);

  // \ref{inner.product}, inner product
  template<class InputIterator1, class InputIterator2, class T>
    T inner_product(InputIterator1 first1, InputIterator1 last1,
                    InputIterator2 first2, T init);
  template<class InputIterator1, class InputIterator2, class T,
           class BinaryOperation1, class BinaryOperation2>
    T inner_product(InputIterator1 first1, InputIterator1 last1,
                    InputIterator2 first2, T init,
                    BinaryOperation1 binary_op1,
                    BinaryOperation2 binary_op2);

  // \ref{transform.reduce}, transform reduce
  template<class InputIterator1, class InputIterator2, class T>
    T transform_reduce(InputIterator1 first1, InputIterator1 last1,
                       InputIterator2 first2,
                       T init);
  template<class InputIterator1, class InputIterator2, class T,
           class BinaryOperation1, class BinaryOperation2>
    T transform_reduce(InputIterator1 first1, InputIterator1 last1,
                       InputIterator2 first2,
                       T init,
                       BinaryOperation1 binary_op1,
                       BinaryOperation2 binary_op2);
  template<class InputIterator, class T,
           class BinaryOperation, class UnaryOperation>
    T transform_reduce(InputIterator first, InputIterator last,
                       T init,
                       BinaryOperation binary_op, UnaryOperation unary_op);
  template<class ExecutionPolicy,
           class ForwardIterator1, class ForwardIterator2, class T>
    T transform_reduce(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                       ForwardIterator1 first1, ForwardIterator1 last1,
                       ForwardIterator2 first2,
                       T init);
  template<class ExecutionPolicy,
           class ForwardIterator1, class ForwardIterator2, class T,
           class BinaryOperation1, class BinaryOperation2>
    T transform_reduce(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                       ForwardIterator1 first1, ForwardIterator1 last1,
                       ForwardIterator2 first2,
                       T init,
                       BinaryOperation1 binary_op1,
                       BinaryOperation2 binary_op2);
  template<class ExecutionPolicy,
           class ForwardIterator, class T,
           class BinaryOperation, class UnaryOperation>
    T transform_reduce(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                       ForwardIterator first, ForwardIterator last,
                       T init,
                       BinaryOperation binary_op, UnaryOperation unary_op);

  // \ref{partial.sum}, partial sum
  template<class InputIterator, class OutputIterator>
    OutputIterator partial_sum(InputIterator first,
                               InputIterator last,
                               OutputIterator result);
  template<class InputIterator, class OutputIterator, class BinaryOperation>
    OutputIterator partial_sum(InputIterator first,
                               InputIterator last,
                               OutputIterator result,
                               BinaryOperation binary_op);

  // \ref{exclusive.scan}, exclusive scan
  template<class InputIterator, class OutputIterator, class T>
    OutputIterator exclusive_scan(InputIterator first, InputIterator last,
                                  OutputIterator result,
                                  T init);
  template<class InputIterator, class OutputIterator, class T, class BinaryOperation>
    OutputIterator exclusive_scan(InputIterator first, InputIterator last,
                                  OutputIterator result,
                                  T init, BinaryOperation binary_op);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2, class T>
    ForwardIterator2 exclusive_scan(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                                    ForwardIterator1 first, ForwardIterator1 last,
                                    ForwardIterator2 result,
                                    T init);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2, class T,
           class BinaryOperation>
    ForwardIterator2 exclusive_scan(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                                    ForwardIterator1 first, ForwardIterator1 last,
                                    ForwardIterator2 result,
                                    T init, BinaryOperation binary_op);

  // \ref{inclusive.scan}, inclusive scan
  template<class InputIterator, class OutputIterator>
    OutputIterator inclusive_scan(InputIterator first, InputIterator last,
                                  OutputIterator result);
  template<class InputIterator, class OutputIterator, class BinaryOperation>
    OutputIterator inclusive_scan(InputIterator first, InputIterator last,
                                  OutputIterator result,
                                  BinaryOperation binary_op);
  template<class InputIterator, class OutputIterator, class BinaryOperation, class T>
    OutputIterator inclusive_scan(InputIterator first, InputIterator last,
                                  OutputIterator result,
                                  BinaryOperation binary_op, T init);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2>
    ForwardIterator2 inclusive_scan(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                                    ForwardIterator1 first, ForwardIterator1 last,
                                    ForwardIterator2 result);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
           class BinaryOperation>
    ForwardIterator2 inclusive_scan(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                                    ForwardIterator1 first, ForwardIterator1 last,
                                    ForwardIterator2 result,
                                    BinaryOperation binary_op);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
           class BinaryOperation, class T>
    ForwardIterator2 inclusive_scan(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                                    ForwardIterator1 first, ForwardIterator1 last,
                                    ForwardIterator2 result,
                                    BinaryOperation binary_op, T init);

  // \ref{transform.exclusive.scan}, transform exclusive scan
  template<class InputIterator, class OutputIterator, class T,
           class BinaryOperation, class UnaryOperation>
    OutputIterator transform_exclusive_scan(InputIterator first, InputIterator last,
                                            OutputIterator result,
                                            T init,
                                            BinaryOperation binary_op,
                                            UnaryOperation unary_op);
  template<class ExecutionPolicy,
           class ForwardIterator1, class ForwardIterator2, class T,
           class BinaryOperation, class UnaryOperation>
    ForwardIterator2 transform_exclusive_scan(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                                              ForwardIterator1 first, ForwardIterator1 last,
                                              ForwardIterator2 result,
                                              T init,
                                              BinaryOperation binary_op,
                                              UnaryOperation unary_op);

  // \ref{transform.inclusive.scan}, transform inclusive scan
  template<class InputIterator, class OutputIterator,
           class BinaryOperation, class UnaryOperation>
    OutputIterator transform_inclusive_scan(InputIterator first, InputIterator last,
                                            OutputIterator result,
                                            BinaryOperation binary_op,
                                            UnaryOperation unary_op);
  template<class InputIterator, class OutputIterator,
           class BinaryOperation, class UnaryOperation, class T>
    OutputIterator transform_inclusive_scan(InputIterator first, InputIterator last,
                                            OutputIterator result,
                                            BinaryOperation binary_op,
                                            UnaryOperation unary_op,
                                            T init);
  template<class ExecutionPolicy,
           class ForwardIterator1, class ForwardIterator2,
           class BinaryOperation, class UnaryOperation>
    ForwardIterator2 transform_inclusive_scan(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                                              ForwardIterator1 first, ForwardIterator1 last,
                                              ForwardIterator2 result,
                                              BinaryOperation binary_op,
                                              UnaryOperation unary_op);
  template<class ExecutionPolicy,
           class ForwardIterator1, class ForwardIterator2,
           class BinaryOperation, class UnaryOperation, class T>
    ForwardIterator2 transform_inclusive_scan(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                                              ForwardIterator1 first, ForwardIterator1 last,
                                              ForwardIterator2 result,
                                              BinaryOperation binary_op,
                                              UnaryOperation unary_op,
                                              T init);

  // \ref{adjacent.difference}, adjacent difference
  template<class InputIterator, class OutputIterator>
    OutputIterator adjacent_difference(InputIterator first,
                                       InputIterator last,
                                       OutputIterator result);
  template<class InputIterator, class OutputIterator, class BinaryOperation>
    OutputIterator adjacent_difference(InputIterator first,
                                       InputIterator last,
                                       OutputIterator result,
                                       BinaryOperation binary_op);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2>
    ForwardIterator2 adjacent_difference(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                                         ForwardIterator1 first,
                                         ForwardIterator1 last,
                                         ForwardIterator2 result);
  template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
           class BinaryOperation>
    ForwardIterator2 adjacent_difference(ExecutionPolicy&& exec, // see \ref{algorithms.parallel.overloads}
                                         ForwardIterator1 first,
                                         ForwardIterator1 last,
                                         ForwardIterator2 result,
                                         BinaryOperation binary_op);

  // \ref{numeric.iota}, iota
  template<class ForwardIterator, class T>
    void iota(ForwardIterator first, ForwardIterator last, T value);

  // \ref{numeric.ops.gcd}, greatest common divisor
  template<class M, class N>
    constexpr common_type_t<M,N> gcd(M m, N n);

  // \ref{numeric.ops.lcm}, least common multiple
  template<class M, class N>
    constexpr common_type_t<M,N> lcm(M m, N n);

  // \ref{numeric.ops.midpoint}, midpoint
  template<class T>
    constexpr T midpoint(T a, T b) noexcept;
  template<class T>
    constexpr T* midpoint(T* a, T* b);
}
\end{codeblock}

\rSec1[numeric.ops]{Generalized numeric operations}

\pnum
\begin{note}
The use of closed ranges as well as semi-open ranges
to specify requirements throughout this subclause is intentional.
\end{note}

\rSec2[numerics.defns]{Definitions}

\indexlibrary{generalized_noncommutative_sum@\tcode{\placeholder{GENERALIZED_NONCOMMUTATIVE_SUM}}}%
\pnum
Define \tcode{\placeholdernc{GENERALIZED_NONCOMMUTATIVE_SUM}(op, a1, ..., aN)}
as follows:
\begin{itemize}
\item
\tcode{a1} when \tcode{N} is \tcode{1}, otherwise

\item
\tcode{op(\placeholdernc{GENERALIZED_NONCOMMUTATIVE_SUM}(op, a1, ..., aK),} \\
\tcode{\phantom{op(}\placeholdernc{GENERALIZED_NONCOMMUTATIVE_SUM}(op, aM, ..., aN))}
for any \tcode{K} where $1 < \mathtt{K}+1 = \mathtt{M} \leq \mathtt{N}$.
\end{itemize}

\indexlibrary{generalized_sum@\tcode{\placeholder{GENERALIZED_SUM}}}%
\pnum
Define \tcode{\placeholdernc{GENERALIZED_SUM}(op, a1, ..., aN)} as
\tcode{\placeholdernc{GENERALIZED_NONCOMMUTATIVE_SUM}(op, b1, ..., bN)},
where
\tcode{b1, ..., bN} may be any permutation of \tcode{a1, ..., aN}.

\rSec2[accumulate]{Accumulate}

\indexlibrary{\idxcode{accumulate}}%
\begin{itemdecl}
template<class InputIterator, class T>
  T accumulate(InputIterator first, InputIterator last, T init);
template<class InputIterator, class T, class BinaryOperation>
  T accumulate(InputIterator first, InputIterator last, T init,
               BinaryOperation binary_op);
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
\tcode{T} shall satisfy
the \oldconcept{CopyConstructible} (\tref{cpp17.copyconstructible})
and \oldconcept{CopyAssignable} (\tref{cpp17.copyassignable}) requirements.
In the range \crange{first}{last},
\tcode{binary_op} shall neither modify elements
nor invalidate iterators or subranges.%
\footnote{The use of fully closed ranges is intentional.}

\pnum
\effects
Computes its result by
initializing the accumulator \tcode{acc} with the initial value \tcode{init}
and then modifies it with
\tcode{acc = std::move(acc) + *i} or
\tcode{acc = binary_op(std::move(acc), *i)}
for every iterator \tcode{i} in the range \range{first}{last} in order.%
\footnote{\tcode{accumulate} is similar
to the APL reduction operator and Common Lisp reduce function,
but it avoids the difficulty of defining the result of reduction
on an empty sequence by always requiring an initial value.}
\end{itemdescr}

\rSec2[reduce]{Reduce}

\indexlibrary{\idxcode{reduce}}%
\begin{itemdecl}
template<class InputIterator>
  typename iterator_traits<InputIterator>::value_type
    reduce(InputIterator first, InputIterator last);
\end{itemdecl}

\begin{itemdescr}
\pnum
\effects Equivalent to:
\begin{codeblock}
return reduce(first, last,
              typename iterator_traits<InputIterator>::value_type{});
\end{codeblock}
\end{itemdescr}

\indexlibrary{\idxcode{reduce}}%
\begin{itemdecl}
template<class ExecutionPolicy, class ForwardIterator>
  typename iterator_traits<ForwardIterator>::value_type
    reduce(ExecutionPolicy&& exec,
           ForwardIterator first, ForwardIterator last);
\end{itemdecl}

\begin{itemdescr}
\pnum
\effects Equivalent to:
\begin{codeblock}
return reduce(std::forward<ExecutionPolicy>(exec), first, last,
              typename iterator_traits<ForwardIterator>::value_type{});
\end{codeblock}
\end{itemdescr}


\indexlibrary{\idxcode{reduce}}%
\begin{itemdecl}
template<class InputIterator, class T>
  T reduce(InputIterator first, InputIterator last, T init);
\end{itemdecl}

\begin{itemdescr}
\pnum
\effects Equivalent to:
\begin{codeblock}
return reduce(first, last, init, plus<>());
\end{codeblock}
\end{itemdescr}

\indexlibrary{\idxcode{reduce}}%
\begin{itemdecl}
template<class ExecutionPolicy, class ForwardIterator, class T>
  T reduce(ExecutionPolicy&& exec,
           ForwardIterator first, ForwardIterator last, T init);
\end{itemdecl}

\begin{itemdescr}
\pnum
\effects Equivalent to:
\begin{codeblock}
return reduce(std::forward<ExecutionPolicy>(exec), first, last, init, plus<>());
\end{codeblock}
\end{itemdescr}


\indexlibrary{\idxcode{reduce}}%
\begin{itemdecl}
template<class InputIterator, class T, class BinaryOperation>
  T reduce(InputIterator first, InputIterator last, T init,
           BinaryOperation binary_op);
template<class ExecutionPolicy, class ForwardIterator, class T, class BinaryOperation>
  T reduce(ExecutionPolicy&& exec,
           ForwardIterator first, ForwardIterator last, T init,
           BinaryOperation binary_op);
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
\begin{itemize}
\item
  \tcode{T} shall be \oldconcept{MoveConstructible} (\tref{cpp17.moveconstructible}).
\item
  All of \tcode{binary_op(init, *first)}, \tcode{binary_op(*first, init)},
  \tcode{binary_op(init, init)}, and \tcode{binary_op(*first, *first)}
  shall be convertible to \tcode{T}.
\item
  \tcode{binary_op} shall neither invalidate iterators or subranges,
  nor modify elements in the range \crange{first}{last}.
\end{itemize}

\pnum
\returns
\tcode{\placeholdernc{GENERALIZED_SUM}(binary_op, init, *i, ...)}
for every \tcode{i} in \range{first}{last}.

\pnum
\complexity
\bigoh{\tcode{last - first}} applications of \tcode{binary_op}.

\pnum
\begin{note}
The difference between \tcode{reduce} and \tcode{accumulate} is that
\tcode{reduce} applies \tcode{binary_op} in an unspecified order,
which yields a nondeterministic result
for non-associative or non-commutative \tcode{binary_op}
such as floating-point addition.
\end{note}
\end{itemdescr}

\rSec2[inner.product]{Inner product}

\indexlibrary{\idxcode{inner_product}}%
\begin{itemdecl}
template<class InputIterator1, class InputIterator2, class T>
  T inner_product(InputIterator1 first1, InputIterator1 last1,
                  InputIterator2 first2, T init);
template<class InputIterator1, class InputIterator2, class T,
         class BinaryOperation1, class BinaryOperation2>
  T inner_product(InputIterator1 first1, InputIterator1 last1,
                  InputIterator2 first2, T init,
                  BinaryOperation1 binary_op1,
                  BinaryOperation2 binary_op2);
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
\tcode{T} shall satisfy
the \oldconcept{CopyConstructible} (\tref{cpp17.copyconstructible})
and \oldconcept{CopyAssignable} (\tref{cpp17.copyassignable}) requirements.
In the ranges \crange{first1}{last1} and
\crange{first2}{first2 + (last1 - first1)}
\tcode{binary_op1} and \tcode{binary_op2}
shall neither modify elements nor invalidate iterators or subranges.%
\footnote{The use of fully closed ranges is intentional.}

\pnum
\effects
Computes its result by
initializing the accumulator \tcode{acc} with the initial value \tcode{init}
and then modifying it with
\tcode{acc = std::move(acc) + (*i1) * (*i2)} or
\tcode{acc = binary_op1(std::move(acc), binary_op2(*i1, *i2))}
for every iterator \tcode{i1} in the range \range{first1}{last1}
and iterator \tcode{i2} in the range \range{first2}{first2 + (last1 - first1)}
in order.
\end{itemdescr}

\rSec2[transform.reduce]{Transform reduce}
\indexlibrary{\idxcode{transform_reduce}}%
\begin{itemdecl}
template<class InputIterator1, class InputIterator2, class T>
  T transform_reduce(InputIterator1 first1, InputIterator1 last1,
                     InputIterator2 first2,
                     T init);
\end{itemdecl}

\begin{itemdescr}
\pnum
\effects
Equivalent to:
\begin{codeblock}
return transform_reduce(first1, last1, first2, init, plus<>(), multiplies<>());
\end{codeblock}
\end{itemdescr}

\indexlibrary{\idxcode{transform_reduce}}%
\begin{itemdecl}
template<class ExecutionPolicy,
         class ForwardIterator1, class ForwardIterator2, class T>
  T transform_reduce(ExecutionPolicy&& exec,
                     ForwardIterator1 first1, ForwardIterator1 last1,
                     ForwardIterator2 first2,
                     T init);
\end{itemdecl}

\begin{itemdescr}
\pnum
\effects
Equivalent to:
\begin{codeblock}
return transform_reduce(std::forward<ExecutionPolicy>(exec),
                        first1, last1, first2, init, plus<>(), multiplies<>());
\end{codeblock}
\end{itemdescr}

\indexlibrary{\idxcode{transform_reduce}}%
\begin{itemdecl}
template<class InputIterator1, class InputIterator2, class T,
         class BinaryOperation1, class BinaryOperation2>
  T transform_reduce(InputIterator1 first1, InputIterator1 last1,
                     InputIterator2 first2,
                     T init,
                     BinaryOperation1 binary_op1,
                     BinaryOperation2 binary_op2);
template<class ExecutionPolicy,
         class ForwardIterator1, class ForwardIterator2, class T,
         class BinaryOperation1, class BinaryOperation2>
  T transform_reduce(ExecutionPolicy&& exec,
                     ForwardIterator1 first1, ForwardIterator1 last1,
                     ForwardIterator2 first2,
                     T init,
                     BinaryOperation1 binary_op1,
                     BinaryOperation2 binary_op2);
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
\begin{itemize}
\item
  \tcode{T} shall be \oldconcept{MoveConstructible} (\tref{cpp17.moveconstructible}).
\item
  All of
  \begin{itemize}
  \item \tcode{binary_op1(init, init)},
  \item \tcode{binary_op1(init, binary_op2(*first1, *first2))},
  \item \tcode{binary_op1(binary_op2(*first1, *first2), init)}, and
  \item \tcode{binary_op1(binary_op2(*first1, *first2), binary_op2(*first1, *first2))}
  \end{itemize}
  shall be convertible to \tcode{T}.
\item
  Neither \tcode{binary_op1} nor \tcode{binary_op2}
  shall invalidate subranges, nor modify elements in the ranges
  \crange{first1}{last1} and \crange{first2}{first2 + (last1 - first1)}.
\end{itemize}

\pnum
\returns
\begin{codeblock}
@\placeholdernc{GENERALIZED_SUM}@(binary_op1, init, binary_op2(*i, *(first2 + (i - first1))), ...)
\end{codeblock}
for every iterator \tcode{i} in \range{first1}{last1}.

\pnum
\complexity
\bigoh{\tcode{last1 - first1}} applications each
of \tcode{binary_op1} and \tcode{binary_op2}.
\end{itemdescr}

\indexlibrary{\idxcode{transform_reduce}}%
\begin{itemdecl}
template<class InputIterator, class T,
         class BinaryOperation, class UnaryOperation>
  T transform_reduce(InputIterator first, InputIterator last, T init,
                     BinaryOperation binary_op, UnaryOperation unary_op);
template<class ExecutionPolicy,
         class ForwardIterator, class T,
         class BinaryOperation, class UnaryOperation>
  T transform_reduce(ExecutionPolicy&& exec,
                     ForwardIterator first, ForwardIterator last,
                     T init, BinaryOperation binary_op, UnaryOperation unary_op);
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
\begin{itemize}
\item
  \tcode{T} shall be \oldconcept{MoveConstructible} (\tref{cpp17.moveconstructible}).
\item
  All of
  \begin{itemize}
  \item \tcode{binary_op(init, init)},
  \item \tcode{binary_op(init, unary_op(*first))},
  \item \tcode{binary_op(unary_op(*first), init)}, and
  \item \tcode{binary_op(unary_op(*first), unary_op(*first))}
  \end{itemize}
  shall be convertible to \tcode{T}.
\item
  Neither \tcode{unary_op} nor \tcode{binary_op} shall invalidate subranges,
  nor modify elements in the range \crange{first}{last}.
\end{itemize}

\pnum
\returns
\begin{codeblock}
@\placeholdernc{GENERALIZED_SUM}@(binary_op, init, unary_op(*i), ...)
\end{codeblock}
for every iterator \tcode{i} in \range{first}{last}.

\pnum
\complexity
\bigoh{\tcode{last - first}} applications each of \tcode{unary_op} and
\tcode{binary_op}.

\pnum
\begin{note}
\tcode{transform_reduce} does not apply \tcode{unary_op} to \tcode{init}.
\end{note}
\end{itemdescr}

\rSec2[partial.sum]{Partial sum}

\indexlibrary{\idxcode{partial_sum}}%
\begin{itemdecl}
template<class InputIterator, class OutputIterator>
  OutputIterator partial_sum(
    InputIterator first, InputIterator last,
    OutputIterator result);
template<class InputIterator, class OutputIterator, class BinaryOperation>
  OutputIterator partial_sum(
    InputIterator first, InputIterator last,
    OutputIterator result, BinaryOperation binary_op);
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
\tcode{InputIterator}'s value type shall be constructible
from the type of \tcode{*first}.
The result of the
expression \tcode{std::move(acc) + *i} or \tcode{binary_op(std::move(acc), *i)}
shall be implicitly convertible to \tcode{InputIterator}'s value type.
\tcode{acc} shall be writable\iref{iterator.requirements.general} to
the \tcode{result} output iterator.
In the ranges \crange{first}{last} and \crange{result}{result + (last - first)}
\tcode{binary_op} shall neither modify elements
nor invalidate iterators or subranges.%
\footnote{The use of fully closed ranges is intentional.}

\pnum
\effects
For a non-empty range,
the function creates an accumulator \tcode{acc}
whose type is \tcode{InputIterator}'s value type,
initializes it with \tcode{*first},
and assigns the result to \tcode{*result}.
For every iterator \tcode{i} in \range{first + 1}{last} in order,
\tcode{acc} is then modified by
\tcode{acc = std::move(acc) + *i} or \tcode{acc = binary_op(std::move(acc), *i)}
and the result is assigned to \tcode{*(result + (i - first))}.

\pnum
\returns
\tcode{result + (last - first)}.

\pnum
\complexity
Exactly \tcode{(last - first) - 1} applications of the binary operation.

\pnum
\remarks
\tcode{result} may be equal to \tcode{first}.
\end{itemdescr}

\rSec2[exclusive.scan]{Exclusive scan}

\indexlibrary{\idxcode{exclusive_scan}}%
\begin{itemdecl}
template<class InputIterator, class OutputIterator, class T>
  OutputIterator exclusive_scan(InputIterator first, InputIterator last,
                                OutputIterator result, T init);
\end{itemdecl}

\begin{itemdescr}
\pnum
\effects Equivalent to:
\begin{codeblock}
return exclusive_scan(first, last, result, init, plus<>());
\end{codeblock}
\end{itemdescr}

\indexlibrary{\idxcode{exclusive_scan}}%
\begin{itemdecl}
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2, class T>
  ForwardIterator2 exclusive_scan(ExecutionPolicy&& exec,
                                  ForwardIterator1 first, ForwardIterator1 last,
                                  ForwardIterator2 result, T init);
\end{itemdecl}

\begin{itemdescr}
\pnum
\effects Equivalent to:
\begin{codeblock}
return exclusive_scan(std::forward<ExecutionPolicy>(exec),
                      first, last, result, init, plus<>());
\end{codeblock}
\end{itemdescr}

\indexlibrary{\idxcode{exclusive_scan}}%
\begin{itemdecl}
template<class InputIterator, class OutputIterator, class T, class BinaryOperation>
  OutputIterator exclusive_scan(InputIterator first, InputIterator last,
                                OutputIterator result, T init, BinaryOperation binary_op);
template<class ExecutionPolicy,
         class ForwardIterator1, class ForwardIterator2, class T, class BinaryOperation>
  ForwardIterator2 exclusive_scan(ExecutionPolicy&& exec,
                                  ForwardIterator1 first, ForwardIterator1 last,
                                  ForwardIterator2 result, T init, BinaryOperation binary_op);
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
\begin{itemize}
\item
  \tcode{T} shall be \oldconcept{MoveConstructible} (\tref{cpp17.moveconstructible}).
\item
  All of
  \tcode{binary_op(init, init)},
  \tcode{binary_op(init, *first)},
  and \tcode{binary_op(*first, *first)}
  shall be convertible to \tcode{T}.
\item
  \tcode{binary_op} shall neither invalidate iterators or subranges,
  nor modify elements in
  the ranges \crange{first}{last} or \crange{result}{result + (last - first)}.
\end{itemize}

\pnum
\effects
For each integer \tcode{K} in \range{0}{last - first}
assigns through \tcode{result + K} the value of:
\begin{codeblock}
@\placeholdernc{GENERALIZED_NONCOMMUTATIVE_SUM}@(
    binary_op, init, *(first + 0), *(first + 1), ..., *(first + K - 1))
\end{codeblock}

\pnum
\returns
The end of the resulting range beginning at \tcode{result}.

\pnum
\complexity
\bigoh{\tcode{last - first}} applications of \tcode{binary_op}.

\pnum
\remarks
\tcode{result} may be equal to \tcode{first}.

\pnum
\begin{note}
The difference between \tcode{exclusive_scan} and \tcode{inclusive_scan} is
that \tcode{exclusive_scan} excludes the $i^\text{th}$ input element
from the $i^\text{th}$ sum.
If \tcode{binary_op} is not mathematically associative,
the behavior of \tcode{exclusive_scan} may be nondeterministic.
\end{note}
\end{itemdescr}

\rSec2[inclusive.scan]{Inclusive scan}

\indexlibrary{\idxcode{inclusive_scan}}%
\begin{itemdecl}
template<class InputIterator, class OutputIterator>
  OutputIterator inclusive_scan(InputIterator first, InputIterator last, OutputIterator result);
\end{itemdecl}

\begin{itemdescr}
\pnum
\effects Equivalent to:
\begin{codeblock}
return inclusive_scan(first, last, result, plus<>());
\end{codeblock}
\end{itemdescr}

\indexlibrary{\idxcode{inclusive_scan}}%
\begin{itemdecl}
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2>
  ForwardIterator2 inclusive_scan(ExecutionPolicy&& exec,
                                ForwardIterator1 first, ForwardIterator1 last,
                                ForwardIterator2 result);
\end{itemdecl}

\begin{itemdescr}
\pnum
\effects Equivalent to:
\begin{codeblock}
return inclusive_scan(std::forward<ExecutionPolicy>(exec), first, last, result, plus<>());
\end{codeblock}
\end{itemdescr}

\indexlibrary{\idxcode{inclusive_scan}}%
\begin{itemdecl}
template<class InputIterator, class OutputIterator, class BinaryOperation>
  OutputIterator inclusive_scan(InputIterator first, InputIterator last,
                                OutputIterator result, BinaryOperation binary_op);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
         class BinaryOperation>
  ForwardIterator2 inclusive_scan(ExecutionPolicy&& exec,
                                  ForwardIterator1 first, ForwardIterator1 last,
                                  ForwardIterator2 result, BinaryOperation binary_op);

template<class InputIterator, class OutputIterator, class BinaryOperation, class T>
  OutputIterator inclusive_scan(InputIterator first, InputIterator last,
                                OutputIterator result, BinaryOperation binary_op, T init);
template<class ExecutionPolicy,
         class ForwardIterator1, class ForwardIterator2, class BinaryOperation, class T>
  ForwardIterator2 inclusive_scan(ExecutionPolicy&& exec,
                                  ForwardIterator1 first, ForwardIterator1 last,
                                  ForwardIterator2 result, BinaryOperation binary_op, T init);
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
\begin{itemize}
\item
  If \tcode{init} is provided,
  \tcode{T} shall be \oldconcept{MoveConstructible} (\tref{cpp17.moveconstructible});
  otherwise, \tcode{ForwardIterator1}'s value type
  shall be \oldconcept{MoveConstructible}.
\item
  If \tcode{init} is provided, all of
  \tcode{binary_op(init, init)},
  \tcode{binary_op(init, *first)}, and
  \tcode{binary_op(*first, *first)}
  shall be convertible to \tcode{T};
  otherwise, \tcode{binary_op(*first, *first)}
  shall be convertible to \tcode{ForwardIterator1}'s value type.
\item
  \tcode{binary_op} shall neither invalidate iterators or subranges,
  nor modify elements in
  the ranges \crange{first}{last} or \crange{result}{result + (last - first)}.
\end{itemize}

\pnum
\effects
For each integer \tcode{K} in \range{0}{last - first}
assigns through \tcode{result + K} the value of
\begin{itemize}
\item
  \tcode{\placeholdernc{GENERALIZED_NONCOMMUTATIVE_SUM}(\\\phantom{\tcode{\ \ \ \ }}binary_op,
  init, *(first + 0), *(first + 1), ..., *(first + K))}\\if \tcode{init} is provided, or
\item
  \tcode{\placeholdernc{GENERALIZED_NONCOMMUTATIVE_SUM}(\\\phantom{\tcode{\ \ \ \ }}binary_op,
  *(first + 0), *(first + 1), ..., *(first + K))}\\otherwise.
\end{itemize}

\pnum
\returns
The end of the resulting range beginning at \tcode{result}.

\pnum
\complexity
\bigoh{\tcode{last - first}} applications of \tcode{binary_op}.

\pnum
\remarks
\tcode{result} may be equal to \tcode{first}.

\pnum
\begin{note}
The difference between \tcode{exclusive_scan} and \tcode{inclusive_scan} is
that \tcode{inclusive_scan} includes the $i^\text{th}$ input element
in the $i^\text{th}$ sum.
If \tcode{binary_op} is not mathematically associative,
the behavior of \tcode{inclusive_scan} may be nondeterministic.
\end{note}
\end{itemdescr}

\rSec2[transform.exclusive.scan]{Transform exclusive scan}

\indexlibrary{\idxcode{transform_exclusive_scan}}%
\begin{itemdecl}
template<class InputIterator, class OutputIterator, class T,
         class BinaryOperation, class UnaryOperation>
  OutputIterator transform_exclusive_scan(InputIterator first, InputIterator last,
                                          OutputIterator result, T init,
                                          BinaryOperation binary_op,UnaryOperation unary_op);
template<class ExecutionPolicy,
         class ForwardIterator1, class ForwardIterator2, class T,
         class BinaryOperation, class UnaryOperation>
  ForwardIterator2 transform_exclusive_scan(ExecutionPolicy&& exec,
                                            ForwardIterator1 first, ForwardIterator1 last,
                                            ForwardIterator2 result, T init,
                                            BinaryOperation binary_op, UnaryOperation unary_op);
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
\begin{itemize}
\item
  \tcode{T} shall be \oldconcept{MoveConstructible} (\tref{cpp17.moveconstructible}).
\item
  All of
  \begin{itemize}
  \item \tcode{binary_op(init, init)},
  \item \tcode{binary_op(init, unary_op(*first))}, and
  \item \tcode{binary_op(unary_op(*first), unary_op(*first))}
  \end{itemize}
  shall be convertible to \tcode{T}.
\item
  Neither \tcode{unary_op} nor \tcode{binary_op} shall
  invalidate iterators or subranges, nor modify elements in
  the ranges \crange{first}{last} or \crange{result}{result + (last - first)}.
\end{itemize}

\pnum
\effects
For each integer \tcode{K} in \range{0}{last - first}
assigns through \tcode{result + K} the value of:
\begin{codeblock}
@\placeholdernc{GENERALIZED_NONCOMMUTATIVE_SUM}@(
    binary_op, init,
    unary_op(*(first + 0)), unary_op(*(first + 1)), ..., unary_op(*(first + K - 1)))
\end{codeblock}

\pnum
\returns
The end of the resulting range beginning at \tcode{result}.

\pnum
\complexity
\bigoh{\tcode{last - first}} applications each
of \tcode{unary_op} and \tcode{binary_op}.

\pnum
\remarks
\tcode{result} may be equal to \tcode{first}.

\pnum
\begin{note}
The difference between \tcode{transform_exclusive_scan} and
\tcode{transform_inclusive_scan} is that \tcode{transform_exclusive_scan}
excludes the $i^\text{th}$ input element from the $i^\text{th}$ sum.
If \tcode{binary_op} is not mathematically associative,
the behavior of \tcode{transform_exclusive_scan} may be nondeterministic.
\tcode{transform_exclusive_scan}
does not apply \tcode{unary_op} to \tcode{init}.
\end{note}
\end{itemdescr}

\rSec2[transform.inclusive.scan]{Transform inclusive scan}

\indexlibrary{\idxcode{transform_inclusive_scan}}%
\begin{itemdecl}
template<class InputIterator, class OutputIterator,
         class BinaryOperation, class UnaryOperation>
  OutputIterator transform_inclusive_scan(InputIterator first, InputIterator last,
                                          OutputIterator result,
                                          BinaryOperation binary_op, UnaryOperation unary_op);
template<class ExecutionPolicy,
         class ForwardIterator1, class ForwardIterator2,
         class BinaryOperation, class UnaryOperation>
  ForwardIterator2 transform_inclusive_scan(ExecutionPolicy&& exec,
                                            ForwardIterator1 first, ForwardIterator1 last,
                                            ForwardIterator2 result,
                                            BinaryOperation binary_op, UnaryOperation unary_op);
template<class InputIterator, class OutputIterator,
         class BinaryOperation, class UnaryOperation, class T>
  OutputIterator transform_inclusive_scan(InputIterator first, InputIterator last,
                                          OutputIterator result,
                                          BinaryOperation binary_op, UnaryOperation unary_op,
                                          T init);
template<class ExecutionPolicy,
         class ForwardIterator1, class ForwardIterator2,
         class BinaryOperation, class UnaryOperation, class T>
  ForwardIterator2 transform_inclusive_scan(ExecutionPolicy&& exec,
                                            ForwardIterator1 first, ForwardIterator1 last,
                                            ForwardIterator2 result,
                                            BinaryOperation binary_op, UnaryOperation unary_op,
                                            T init);
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
\begin{itemize}
\item
  If \tcode{init} is provided, \tcode{T} shall be
  \oldconcept{MoveConstructible} (\tref{cpp17.moveconstructible});
  otherwise, \tcode{ForwardIterator1}'s value type shall be
  \oldconcept{MoveConstructible}.
\item
  If \tcode{init} is provided, all of
  \begin{itemize}
  \item \tcode{binary_op(init, init)},
  \item \tcode{binary_op(init, unary_op(*first))}, and
  \item \tcode{binary_op(unary_op(*first), unary_op(*first))}
  \end{itemize}
  shall be convertible to \tcode{T};
  otherwise, \tcode{binary_op(unary_op(*first), unary_op(*first))}
  shall be convertible to \tcode{ForwardIterator1}'s value type.
\item
  Neither \tcode{unary_op} nor \tcode{binary_op} shall
  invalidate iterators or subranges, nor modify elements in
  the ranges \crange{first}{last} or \crange{result}{result + (last - first)}.
\end{itemize}

\pnum
\effects
For each integer \tcode{K} in \range{0}{last - first}
assigns through \tcode{result + K} the value of
\begin{itemize}
\item
  \tcode{\placeholdernc{GENERALIZED_NONCOMMUTATIVE_SUM}(\\\phantom{\tcode{\ \ \ \ }}binary_op, init,\\\phantom{\tcode{\ \ \ \ }}unary_op(*(first + 0)), unary_op(*(first + 1)), ..., unary_op(*(first + K)))}\\
  if \tcode{init} is provided, or
\item
  \tcode{\placeholdernc{GENERALIZED_NONCOMMUTATIVE_SUM}(\\\phantom{\tcode{\ \ \ \ }}binary_op,\\\phantom{\tcode{\ \ \ \ }}unary_op(*(first + 0)), unary_op(*(first + 1)), ..., unary_op(*(first + K)))}\\
  otherwise.
\end{itemize}

\pnum
\returns
The end of the resulting range beginning at \tcode{result}.

\pnum
\complexity
\bigoh{\tcode{last - first}} applications each
of \tcode{unary_op} and \tcode{binary_op}.

\pnum
\remarks
\tcode{result} may be equal to \tcode{first}.

\pnum
\begin{note}
The difference between \tcode{transform_exclusive_scan} and
\tcode{transform_inclusive_scan} is that \tcode{transform_inclusive_scan}
includes the $i^\text{th}$ input element in the $i^\text{th}$ sum.
If \tcode{binary_op} is not mathematically associative,
the behavior of \tcode{transform_inclusive_scan} may be nondeterministic.
\tcode{transform_inclusive_scan} does not
apply \tcode{unary_op} to \tcode{init}.
\end{note}
\end{itemdescr}

\rSec2[adjacent.difference]{Adjacent difference}

\indexlibrary{\idxcode{adjacent_difference}}%
\begin{itemdecl}
template<class InputIterator, class OutputIterator>
  OutputIterator
    adjacent_difference(InputIterator first, InputIterator last, OutputIterator result);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2>
  ForwardIterator2
    adjacent_difference(ExecutionPolicy&& exec,
                        ForwardIterator1 first, ForwardIterator1 last, ForwardIterator2 result);

template<class InputIterator, class OutputIterator, class BinaryOperation>
  OutputIterator
    adjacent_difference(InputIterator first, InputIterator last,
                        OutputIterator result, BinaryOperation binary_op);
template<class ExecutionPolicy, class ForwardIterator1, class ForwardIterator2,
         class BinaryOperation>
  ForwardIterator2
    adjacent_difference(ExecutionPolicy&& exec,
                        ForwardIterator1 first, ForwardIterator1 last,
                        ForwardIterator2 result, BinaryOperation binary_op);
\end{itemdecl}

\begin{itemdescr}
\pnum
Let \tcode{T} be the value type of \tcode{decltype(first)}.
For the overloads that do not take an argument \tcode{binary_op},
let \tcode{binary_op} be an lvalue
that denotes an object of type \tcode{minus<>}.

\pnum
\requires
\begin{itemize}
\item
  For the overloads with no \tcode{ExecutionPolicy},
  \tcode{T} shall be \oldconcept{MoveAssignable} (\tref{cpp17.moveassignable}) and
  shall be constructible from the type of \tcode{*first}.
  \tcode{acc} (defined below) shall be
  writable\iref{iterator.requirements.general}
  to the \tcode{result} output iterator.
  The result of the expression \tcode{binary_op(val, std::move(acc))}
  shall be writable to the \tcode{result} output iterator.
\item
  For the overloads with an \tcode{ExecutionPolicy},
  the result of the expressions \tcode{binary_op(*first, *first)} and
  \tcode{*first} shall be writable to \tcode{result}.
\item
  For all overloads, in the ranges \crange{first}{last}
  and \crange{result}{result + (last - first)},
  \tcode{binary_op} shall neither modify elements
  nor invalidate iterators or subranges.%
  \footnote{The use of fully closed ranges is intentional.}
\end{itemize}

\pnum
\effects
For the overloads with no \tcode{ExecutionPolicy} and a non-empty range,
the function creates an accumulator \tcode{acc} of type \tcode{T},
initializes it with \tcode{*first},
and assigns the result to \tcode{*result}.
For every iterator \tcode{i} in \range{first + 1}{last} in order,
creates an object \tcode{val} whose type is \tcode{T},
initializes it with \tcode{*i},
computes \tcode{binary_op(val, std::move(acc))},
assigns the result to \tcode{*(result + (i - first))}, and
move assigns from \tcode{val} to \tcode{acc}.

\pnum
For the overloads with an \tcode{ExecutionPolicy} and a non-empty range,
performs \tcode{*result = *first}.
Then, for every \tcode{d} in \tcode{[1, last - first - 1]},
performs \tcode{*(result + d) = binary_op(*(first + d), *(first + (d - 1)))}.

\pnum
\returns
\tcode{result + (last - first)}.

\pnum
\complexity
Exactly \tcode{(last - first) - 1} applications of the binary operation.

\pnum
\remarks
For the overloads with no \tcode{ExecutionPolicy},
\tcode{result} may be equal to \tcode{first}.
For the overloads with an \tcode{ExecutionPolicy},
the ranges \range{first}{last} and \range{result}{result + (last - first)}
shall not overlap.
\end{itemdescr}

\rSec2[numeric.iota]{Iota}

\indexlibrary{\idxcode{iota}}%
\begin{itemdecl}
template<class ForwardIterator, class T>
  void iota(ForwardIterator first, ForwardIterator last, T value);
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
\tcode{T} shall be convertible to \tcode{ForwardIterator}'s value type.
The expression \tcode{++val}, where \tcode{val} has type \tcode{T},
shall be well-formed.

\pnum
\effects
For each element referred to by the iterator \tcode{i}
in the range \range{first}{last},
assigns \tcode{*i = value} and increments \tcode{value}
as if by \tcode{++value}.

\pnum
\complexity
Exactly \tcode{last - first} increments and assignments.
\end{itemdescr}

\rSec2[numeric.ops.gcd]{Greatest common divisor}

\indexlibrary{\idxcode{gcd}}%
\begin{itemdecl}
template<class M, class N>
  constexpr common_type_t<M,N> gcd(M m, N n);
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
$|\tcode{m}|$ and $|\tcode{n}|$
shall be representable as a value of \tcode{common_type_t<M, N>}.
\begin{note}
These requirements ensure, for example,
that $\tcode{gcd(m, m)} = |\tcode{m}|$
is representable as a value of type \tcode{M}.
\end{note}

\pnum
\remarks
If either \tcode{M} or \tcode{N} is not an integer type, or
if either is \cv{}~\tcode{bool}, the program is ill-formed.

\pnum
\returns
Zero when \tcode{m} and \tcode{n} are both zero. Otherwise,
returns the greatest common divisor of $|\tcode{m}|$ and $|\tcode{n}|$.

\pnum
\throws
Nothing.
\end{itemdescr}

\rSec2[numeric.ops.lcm]{Least common multiple}

\indexlibrary{\idxcode{lcm}}%
\begin{itemdecl}
template<class M, class N>
  constexpr common_type_t<M,N> lcm(M m, N n);
\end{itemdecl}

\begin{itemdescr}
\pnum
\requires
$|\tcode{m}|$ and $|\tcode{n}|$
shall be representable as a value of \tcode{common_type_t<M, N>}.
The least common multiple of $|\tcode{m}|$ and $|\tcode{n}|$
shall be representable as a value of type \tcode{common_type_t<M,N>}.

\pnum
\remarks
If either \tcode{M} or \tcode{N} is not an integer type, or
if either is \cv{}~\tcode{bool} the program is ill-formed.

\pnum
\returns
Zero when either \tcode{m} or \tcode{n} is zero.
Otherwise, returns the least common multiple of $|\tcode{m}|$ and $|\tcode{n}|$.

\pnum
\throws
Nothing.
\end{itemdescr}

\rSec2[numeric.ops.midpoint]{Midpoint}

\indexlibrary{\idxcode{midpoint}}%
\begin{itemdecl}
template<class T>
  constexpr T midpoint(T a, T b) noexcept;
\end{itemdecl}
\begin{itemdescr}
\pnum
\constraints
\tcode{T} is an arithmetic type other than \tcode{bool}.

\pnum
\returns
Half the sum of \tcode{a} and \tcode{b}.
If \tcode{T} is an integer type and the sum is odd,
the result is rounded towards \tcode{a}.

\pnum
\remarks
No overflow occurs.
If \tcode{T} is a floating-point type, at most one inexact operation occurs.
\end{itemdescr}

\indexlibrary{\idxcode{midpoint}}%
\begin{itemdecl}
template<class T>
  constexpr T* midpoint(T* a, T* b);
\end{itemdecl}
\begin{itemdescr}
\pnum
\constraints
\tcode{T} is a complete object type.

\pnum
\expects
\tcode{a} and \tcode{b} point to, respectively,
elements $\tcode{x}[i]$ and $\tcode{x}[j]$ of the same array object \tcode{x}.
\begin{note}
An object that is not an array element is considered to belong
to a single-element array for this purpose; see \ref{expr.unary.op}.
A pointer past the last element of an array \tcode{x} of $n$ elements
is considered to be equivalent to a pointer
to a hypothetical element $\tcode{x}[n]$ for this purpose;
see \ref{basic.compound}.
\end{note}

\pnum
\returns
A pointer to $\tcode{x}[i+\frac{j-i}{2}]$,
where the result of the division is truncated towards zero.
\end{itemdescr}

\rSec1[alg.c.library]{C library algorithms}

\pnum
\indexhdr{cstdlib}%
\begin{note}
The header \tcode{<cstdlib>}\iref{cstdlib.syn}
declares the functions described in this subclause.
\end{note}

\indexlibrary{\idxcode{bsearch}}%
\indexlibrary{\idxcode{qsort}}%
\begin{itemdecl}
void* bsearch(const void* key, const void* base, size_t nmemb, size_t size,
              @\placeholder{c-compare-pred}@* compar);
void* bsearch(const void* key, const void* base, size_t nmemb, size_t size,
              @\placeholder{compare-pred}@* compar);
void qsort(void* base, size_t nmemb, size_t size, @\placeholder{c-compare-pred}@* compar);
void qsort(void* base, size_t nmemb, size_t size, @\placeholder{compare-pred}@* compar);
\end{itemdecl}

\begin{itemdescr}
\pnum
\effects
These functions have the semantics specified in the C standard library.

\pnum
\remarks
The behavior is undefined
unless the objects in the array pointed to by \tcode{base} are of trivial type.

\pnum
\throws
Any exception thrown by \tcode{compar()}\iref{res.on.exception.handling}.
\end{itemdescr}

\xref
ISO C 7.22.5.