[/
 / Copyright (c) 2012 Marshall Clow
 / Copyright (c) 2021, Alan Freitas
 /
 / Distributed under the Boost Software License, Version 1.0. (See accompanying
 / file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
 /]

[/===============]
[section:operators Operators]
[/===============]

[section Introduction]

The header [@../../../../boost/operators.hpp `<boost/operators.hpp>`] supplies
several sets of class templates in `namespace boost`. These templates define
operators at namespace scope in terms of a minimal number of fundamental
operators provided by the class.

[endsect]

[section:rationale Rationale]

Overloaded operators for class types typically occur in groups. If you
can write `x + y`, you probably also want to be able to write `x += y`.
If you can write `x < y,` you also want `x > y`, `x >= y,` and `x <= y`.

Moreover, unless your class has really surprising behavior, some of these
related operators can be defined in terms of others (e.g. `x >= y`
is equivalent to `!(x < y)`).

Replicating this boilerplate for multiple classes is both tedious and
error-prone. The [@../../../../boost/operators.hpp `<boost/operators.hpp>`] templates
help by generating operators for you at namespace scope based on other
operators you have defined in your class.

If, for example, you declare a class like this:

```
class MyInt
    : boost::__operators__<MyInt>
{
    bool operator<(const MyInt& x) const;
    bool operator==(const MyInt& x) const;
    MyInt& operator+=(const MyInt& x);
    MyInt& operator-=(const MyInt& x);
    MyInt& operator*=(const MyInt& x);
    MyInt& operator/=(const MyInt& x);
    MyInt& operator%=(const MyInt& x);
    MyInt& operator|=(const MyInt& x);
    MyInt& operator&=(const MyInt& x);
    MyInt& operator^=(const MyInt& x);
    MyInt& operator++();
    MyInt& operator--();
};
```

then the __operators__<> template adds more than a dozen additional operators, such as
`operator>`, `operator<=`, `operator>=`, and the binary `operator+`.

[link utility.utilities.operators.usage.two_arg Two-argument forms] of the templates are also provided to allow interaction with other types.

[#utility.utilities.operators.rationale.semantics]
This is a ['Summary of Template Semantics]:

# Each operator template completes the concept(s) it describes by defining overloaded operators for its target class.
# The name of an operator class template indicates the [link utility.utilities.operators.rationale.concepts_note concept] that its target class will model.
# Usually, the target class uses an instantiation of the operator class template as a base class. Some operator templates support an
  [link utility.utilities.operators.usage.explicit_instantiation alternate method].
# The concept can be compound, i.e. it may represent a common combination of other, simpler concepts.
# Most operator templates require their target class to support operations related to the operators supplied by
  the template. In accordance with widely accepted [@http://www.gotw.ca/gotw/004.htm coding style recommendations], the
  target class is often required to supply the assignment counterpart operator of the concept's "main operator."
  For example, the `addable` template requires `operator+=(T const&)` and in turn supplies `operator+(T const&, T const&)`.

[#utility.utilities.operators.rationale.concepts_note]
['Note on the use of concepts]: The discussed concepts are not necessarily the standard library's
concepts, such as __CopyConstructible__, although some of them could
be; they are what we call ['concepts with a small 'c']. In
particular, they are different from the former ones in that they ['do
not] describe precise semantics of the operators they require to be
defined, except the requirements that (a) the semantics of the operators
grouped in one concept should be consistent (e.g. effects of
evaluating of `a += b` and `a = a + b` expressions should be the
same), and (b) that the return types of the operators should follow
semantics of return types of corresponding operators for built-in types
(e.g. `operator<` should return a type convertible to `bool`, and
`T::operator-=` should return type convertible to `T`). Such "loose"
requirements make `operators` library applicable to broader set of
target classes from different domains, i.e. eventually more useful.

[endsect]

[section:example Example]

This example shows how some of the [link utility.utilities.operators.arithmetic arithmetic
operator templates] can be used with a geometric point class
template.

```
template <class T>
class point    // note: private inheritance is OK here!
    : boost::addable< point<T>          // point + point
    , boost::subtractable< point<T>     // point - point
    , boost::dividable2< point<T>, T    // point / T
    , boost::multipliable2< point<T>, T // point * T, T * point
      > > > >
{
public:
    point(T, T);
    T x() const;
    T y() const;

    point operator+=(const point&);
    // point operator+(point, const point&) automatically
    // generated by addable.

    point operator-=(const point&);
    // point operator-(point, const point&) automatically
    // generated by subtractable.

    point operator*=(T);
    // point operator*(point, const T&) and
    // point operator*(const T&, point) auto-generated
    // by multipliable.

    point operator/=(T);
    // point operator/(point, const T&) auto-generated
    // by dividable.
private:
    T x_;
    T y_;
};

// now use the point<> class:
template <class T>
T length(const point<T> p)
{
    return sqrt(p.x()*p.x() + p.y()*p.y());
}

const point<float> right(0, 1);
const point<float> up(1, 0);
const point<float> pi_over_4 = up + right;
const point<float> pi_over_4_normalized = pi_over_4 / length(pi_over_4);
```

[endsect]

[section:usage Usage]

[section:two_arg Two-Argument Template Forms]

The arguments to a binary operator commonly have identical types, but
it is not unusual to want to define operators which combine different
types. For [link utility.utilities.operators.example example], one might want to multiply a
mathematical vector by a scalar. The two-argument template forms of the
arithmetic operator templates are supplied for this purpose. When
applying the two-argument form of a template, the desired return type of
the operators typically determines which of the two types in question
should be derived from the operator template.

For example, if the result of `T + U` is of type `T`, then `T` (not `U`)
should be derived from [link utility.utilities.operators.arithmetic.smpl_oprs.addable2 `addable<T, U>`]. The comparison
templates [link utility.utilities.operators.arithmetic.smpl_oprs.less_than_comparable2 `less_than_comparable<T,U>`],
[link utility.utilities.operators.arithmetic.smpl_oprs.equality_comparable2 `equality_comparable<T, U>`],
[link utility.utilities.operators.arithmetic.smpl_oprs.equivalent2 `equivalent<T, U>`], and
[link utility.utilities.operators.arithmetic.smpl_oprs.partially_ordered2 `partially_ordered<T, U>`] are exceptions to
this guideline, since the return type of the operators they define is `bool`.

On compilers which do not support partial specialization, the
two-argument forms must be specified by using the names shown below with
the trailing `'2'`. The single-argument forms with the
trailing `'1'` are provided for symmetry and to enable certain
applications of the [link utility.utilities.operators.usage.chaining base class chaining]
technique.

[#utility.utilities.operators.usage.two_arg.mixed_arithmetics]
['Mixed Arithmetics]: Another application of the
 two-argument template forms is for mixed
arithmetics between a type `T` and a type `U` that
is convertible to `T`. In this case there are two ways where
the two-argument template forms are helpful: one is to provide the
respective signatures for operator overloading, the second is
performance.

With respect to the operator overloading assume e.g. that
`U` is `int`, that `T` is an user-defined unlimited integer type,
and that `double operator-(double, const T&)` exists.

If one wants to compute `int - T` and does not provide
`T operator-(int, const T&)`, the compiler will consider
`double operator-(double, const T&)` to be a better match
than `T operator-(const T&, const T&)`, which will probably
be different from the user's intention.

To define a complete set of operator signatures, additional 'left'
forms of the two-argument template forms are provided
[link utility.utilities.operators.arithmetic.smpl_oprs.subtractable2_left `subtractable2_left<T, U>`],
[link utility.utilities.operators.arithmetic.smpl_oprs.dividable2_left `dividable2_left<T, U>`], and
[link utility.utilities.operators.arithmetic.smpl_oprs.modable2_left `modable2_left<T, U>`] that
define the signatures for non-commutative operators where
 `U` appears on the left hand side (`operator-(const U&, const T&)`,
`operator/(const U&, const T&)`, `operator%(const U&, const T&)`).

With respect to the performance observe that when one uses the single
type binary operator for mixed type arithmetics, the type `U`
argument has to be converted to type `T`. In practice,
however, there are often more efficient implementations of, say
`T::operator-=(const U&)` that avoid unnecessary conversions
from `U` to `T`.

The two-argument template forms of the arithmetic operator create
additional operator interfaces that use these more efficient implementations.
There is, however, no performance gain in the 'left' forms: they still need a
conversion from `U` to `T` and have an implementation equivalent to the code
that would be automatically created by the compiler if it considered the
single type binary operator to be the best match.

[endsect]

[section:chaining Base Class Chaining and Object Size]

Every operator class template, except the [link utility.utilities.operators.arithmetic.ex_oprs arithmetic examples]
and the [link utility.utilities.operators.deref.iterator iterator helpers], has an additional, but optional,
template type parameter `B`. This parameter will be a publicly-derived base class of
the instantiated template. This means it must be a class type. It can be
used to avoid the bloating of object sizes that is commonly associated
with multiple-inheritance from several empty base classes. See the
[link utility.utilities.operators.old_lib_note note for users of older versions] for more
details.

To provide support for a group of operators, use the
`B` parameter to chain operator templates into a single-base
class hierarchy, demostrated in the [link utility.utilities.operators.example usage example].
The technique is also used by the composite operator templates to group
operator definitions. If a chain becomes too long for the compiler to
support, try replacing some of the operator templates with a single
grouped operator template that chains the old templates together; the
length limit only applies to the number of templates directly in the
chain, not those hidden in group templates.

['Caveat]: to chain to a base class which is ['not] a Boost operator
template when using the [link utility.utilities.operators.usage.two_arg single-argument form] of a Boost
operator template, you must specify the operator template with the
trailing `'1'` in its name. Otherwise the library will assume you mean
to define a binary operation combining the class you intend to use as
a base class and the class you're deriving.

[endsect]

[section:explicit_instantiation Separate Explicit Instantiation]

On some compilers (e.g. Borland, GCC) even single-inheritance
seems to cause an increase in object size in some cases. If you are not
defining a class template, you may get better object-size performance by
avoiding derivation altogether, and instead explicitly instantiating the
operator template as follows:

```
class my_class // lose the inheritance...
{
    //...
};

// explicitly instantiate the operators I need.
template struct less_than_comparable<my_class>;
template struct equality_comparable<my_class>;
template struct incrementable<my_class>;
template struct decrementable<my_class>;
template struct addable<my_class,long>;
template struct subtractable<my_class,long>;
```

Note that some operator templates cannot use this workaround and must
be a base class of their primary operand type. Those templates define
operators which must be member functions, and the workaround needs the
operators to be independent `friend` functions. The relevant templates
are:

* [link utility.utilities.operators.deref.dereference.dereferenceable `dereferenceable<>`]
* [link utility.utilities.operators.deref.dereference.indexable `indexable<>`]
* Any composite operator template that includes at least one of the above

As Daniel Krugler pointed out, this technique violates C++11 [sect]14.6.5/2 \[temp.inject\]
and is thus non-portable. The reasoning is, that the operators injected by the instantiation
of e.g. `less_than_comparable<my_class>` can not be found by ADL according to the
rules given by C++11 [sect]3.4.2/2 \[basic.lookup.argdep\], since `my_class` is not
an associated class of `less_than_comparable<my_class>`. Thus only use this technique
if all else fails.

[endsect]

[section:portability Requirement Portability]

Many compilers (e.g. MSVC 6.3, GCC 2.95.2) will not enforce the
requirements in the operator template tables unless the operations which
depend on them are actually used. This is not standard-conforming
behavior. In particular, although it would be convenient to derive all
your classes which need binary operators from the [link utility.utilities.operators.arithmetic.ex_oprs.operators1 `operators<>`]
and [link utility.utilities.operators.arithmetic.ex_oprs.operators2 `operators2<>`] templates, regardless of
whether they implement all the requirements of those templates, this
shortcut is not portable. Even if this currently works with your
compiler, it may not work later.

[endsect]

[endsect]

[section:arithmetic Arithmetic Operators]

The arithmetic operator templates ease the task of creating a custom
numeric type. Given a core set of operators, the templates add related
operators to the numeric class. These operations are like the ones the
standard arithmetic types have, and may include comparisons, adding,
incrementing, logical and bitwise manipulations, etc. Further,
since most numeric types need more than one of these operators, some
templates are provided to combine several of the basic operator templates
in one declaration.

The requirements for the types used to instantiate the simple operator
templates are specified in terms of expressions which must be valid and
the expression's return type. The composite operator templates only list
what other templates they use. The supplied operations and requirements
of the composite operator templates can be inferred from the operations
and requirements of the listed components.

[section:smpl_oprs Simple Arithmetic Operators]

These templates are "simple" since they provide operators based on a
single operation the base type has to provide. They have an additional
optional template parameter `B`, which is not shown, for the
[link utility.utilities.operators.usage.chaining base class chaining] technique.

The primary operand type `T` needs to be of class type,
built-in types are not supported.

[table Notation
    [[Key] [Description]]
    [[`T`] [primary operand type]]
    [[`t,t1`] [values of type `T`]]
    [[`U`] [alternate operand type]]
    [[`u`] [value of type `U`]]
]

[table Simple Arithmetic Operator Template Classes
    [[Template] [Supplied Operations] [Requirements] [Propagates constexpr]]
    [
        [
          [#utility.utilities.operators.arithmetic.smpl_oprs.less_than_comparable1] `less_than_comparable<T>`

          __less_than_comparable1__`<T>`
        ]

        [
         `bool operator>(const T&, const T&)`

         `bool operator<=(const T&, const T&)`

         `bool operator>=(const T&, const T&)`
        ]

        [
         `t < t1`.

         Return convertible to `bool`. See the [link utility.utilities.operators.arithmetic.smpl_oprs.ordering Ordering Note]
        ]

        [
            Since `C++11`, except [@https://developercommunity.visualstudio.com/content/problem/414193/rejects-valid-constexpr-marked-friend-function-def.html MSVC < v19.22]
        ]
      ]

      [
        [
          [#utility.utilities.operators.arithmetic.smpl_oprs.less_than_comparable2]`less_than_comparable<T,U>`

          __less_than_comparable2__`<T, U>`
        ]

        [
         `bool operator<=(const T&, const U&)`

         `bool operator>=(const T&, const U&)`

         `bool operator>(const U&, const T&)`

         `bool operator<(const U&, const T&)`

         `bool operator<=(const U&, const T&)`

         `bool operator>=(const U&, const T&)`
        ]

        [
         `t < u`. `t > u`.

         Returns convertible to `bool`. See the [link utility.utilities.operators.arithmetic.smpl_oprs.ordering Ordering Note].
        ]

        [
           Since `C++11`, except [@https://developercommunity.visualstudio.com/content/problem/414193/rejects-valid-constexpr-marked-friend-function-def.html MSVC < v19.22]
        ]
      ]

      [
        [
          [#utility.utilities.operators.arithmetic.smpl_oprs.equality_comparable1]`equality_comparable<T>`

           __equality_comparable1__`<T>`
        ]

        [
           `bool operator!=(const T&, const T&)`
        ]

        [
         `t == t1`.

         Return convertible to `bool`.
        ]

        [
           Since `C++11`, except [@https://developercommunity.visualstudio.com/content/problem/414193/rejects-valid-constexpr-marked-friend-function-def.html MSVC < v19.22]
        ]
      ]

      [
        [
          [#utility.utilities.operators.arithmetic.smpl_oprs.equality_comparable2]`equality_comparable<T,U>`

          __equality_comparable2__`<T, U>`
        ]

        [
         `bool operator==(const U&, const T&)`

         `bool operator!=(const U&, const T&)`

         `bool operator!=(const T&, const U&)`
        ]

        [
          `t == u`.

          Return convertible to `bool`.
        ]

         [
            Since `C++11`, except [@https://developercommunity.visualstudio.com/content/problem/414193/rejects-valid-constexpr-marked-friend-function-def.html MSVC < v19.22]
         ]
      ]

      [
        [
          [#utility.utilities.operators.arithmetic.smpl_oprs.addable1]`addable<T>`

          `addable1<T>`
        ]

        [
           `T operator+(const T&, const T&)`
        ]

        [
         `T temp(t); temp += t1`.

         Return convertible to `T`. See the [link utility.utilities.operators.arithmetic.smpl_oprs.symmetry Symmetry Note].
        ]

        [No]
      ]

      [
        [
          [#utility.utilities.operators.arithmetic.smpl_oprs.addable2]`addable<T, U>`

          `addable2<T, U>`
        ]

        [
         `T operator+(const T&, const U&)`

         `T operator+(const U&, const T& )`
        ]

        [
         `T temp(t); temp += u`.

         Return convertible to `T`. See the [link utility.utilities.operators.arithmetic.smpl_oprs.symmetry Symmetry Note].
        ]

        [No]
      ]

      [
        [
          [#utility.utilities.operators.arithmetic.smpl_oprs.subtractable1]`subtractable<T>`

           `subtractable1<T>`
        ]

        [
          `T operator-(const T&, const T&)`
        ]

        [
         `T temp(t); temp -= t1`.

         Return convertible to `T`. See the [link utility.utilities.operators.arithmetic.smpl_oprs.symmetry Symmetry Note].
        ]

        [No]
      ]

      [
        [
          [#utility.utilities.operators.arithmetic.smpl_oprs.subtractable2]`subtractable<T,U>`

          `subtractable2<T, U>`
        ]

        [
          `T operator-(const T&, const U&)`
        ]

        [
         `T temp(t); temp -= u`.

         Return convertible to `T`. See the [link utility.utilities.operators.arithmetic.smpl_oprs.symmetry Symmetry Note].
        ]

        [No]
      ]

      [
        [
          [#utility.utilities.operators.arithmetic.smpl_oprs.subtractable2_left]`subtractable2_left<T,U>`
        ]

        [
          `T operator-(const U&, const T&)`
        ]

        [
         `T temp(u); temp -= t`.

         Return convertible to `T`.
        ]

        [No]
      ]

      [
        [
          [#utility.utilities.operators.arithmetic.smpl_oprs.multipliable1]`multipliable<T>`

          `multipliable1<T>`
        ]

        [
          `T operator*(const T&, const T&)`
        ]

        [
         `T temp(t); temp *= t1`.

         Return convertible to `T`. See the [link utility.utilities.operators.arithmetic.smpl_oprs.symmetry Symmetry Note].
        ]

        [No]
      ]

      [
        [
           [#utility.utilities.operators.arithmetic.smpl_oprs.multipliable2]`multipliable<T,U>`

           `multipliable2<T, U>`
        ]

        [
          `T operator*(const T&, const U&)`

          `T operator*(const U&, const T&)`
        ]

        [
         `T temp(t); temp *= u`.

         Return convertible to `T`. See the [link utility.utilities.operators.arithmetic.smpl_oprs.symmetry Symmetry Note].
        ]

        [No]
      ]

      [
        [
          [#utility.utilities.operators.arithmetic.smpl_oprs.dividable1]`dividable<T>`

          `dividable1<T>`
        ]

        [
          `T operator/(const T&, const T&)`
        ]

        [
          `T temp(t); temp /= t1`.

          Return convertible to `T`. See the [link utility.utilities.operators.arithmetic.smpl_oprs.symmetry Symmetry Note].
        ]

        [No]
      ]

      [
        [
          [#utility.utilities.operators.arithmetic.smpl_oprs.dividable2]`dividable<T, U>`

          `dividable2<T, U>`
        ]

        [
          `T operator/(const T&, const U&)`
        ]

        [
          `T temp(t); temp /= u`.

          Return convertible to `T`. See the [link utility.utilities.operators.arithmetic.smpl_oprs.symmetry Symmetry Note].
        ]

        [No]
      ]

      [
        [
          [#utility.utilities.operators.arithmetic.smpl_oprs.dividable2_left]`dividable2_left<T,U>`
        ]

        [
          `T operator/(const U&, const T&)`
        ]

        [
          `T temp(u); temp /= t`.

          Return convertible to `T`.
        ]

        [No]
      ]

      [
        [
          [#utility.utilities.operators.arithmetic.smpl_oprs.modable1]`modable<T>`

         `modable1<T>`
       ]

        [
          `T operator%(const T&, const T&)`
        ]

        [
         `T temp(t); temp %= t1`.

         Return convertible to `T`. See the [link utility.utilities.operators.arithmetic.smpl_oprs.symmetry Symmetry Note].
        ]

        [No]
      ]

      [
        [
          [#utility.utilities.operators.arithmetic.smpl_oprs.modable2]`modable<T, U>`

         `modable2<T, U>`
        ]

        [
          `T operator%(const T&, const U&)`
        ]

        [
          `T temp(t); temp %= u`.

          Return convertible to `T`. See the [link utility.utilities.operators.arithmetic.smpl_oprs.symmetry Symmetry Note].
        ]

        [No]
      ]

      [
        [
          [#utility.utilities.operators.arithmetic.smpl_oprs.modable2_left]`modable2_left<T,U>`
        ]

        [
          `T operator%(const U&, const T&)`
        ]

        [
          `T temp(u); temp %= t`.

          Return convertible to `T`.
        ]

        [No]
      ]

      [
        [
          [#utility.utilities.operators.arithmetic.smpl_oprs.orable1]`orable<T>`

          `orable1<T>`
        ]

        [
           `T operator|(const T&, const T&)`
        ]

        [
          `T temp(t); temp |= t1`.

          Return convertible to `T`. See the [link utility.utilities.operators.arithmetic.smpl_oprs.symmetry Symmetry Note].
        ]

        [No]
      ]

      [
        [
          [#utility.utilities.operators.arithmetic.smpl_oprs.orable2]`orable<T, U>`

          `orable2<T, U>`
        ]

        [
          `T operator|(const T&, const U&)`

          `T operator|(const U&, const T&)`
        ]

        [
          `T temp(t); temp |= u`.

          Return convertible to `T`. See the [link utility.utilities.operators.arithmetic.smpl_oprs.symmetry Symmetry Note].
        ]

        [No]
      ]

      [
        [
           [#utility.utilities.operators.arithmetic.smpl_oprs.andable1]`andable<T>`

           `andable1<T>`
        ]

        [
          `T operator&(const T&, const T&)`
        ]

        [
           `T temp(t); temp &= t1`.

           Return convertible to `T`. See the [link utility.utilities.operators.arithmetic.smpl_oprs.symmetry Symmetry Note].
        ]

        [No]
      ]

      [
        [
          [#utility.utilities.operators.arithmetic.smpl_oprs.andable2]`andable<T, U>`

          `andable2<T, U>`
       ]

        [
          `T operator&(const T&, const U&)`

          `T operator&(const U&, const T&)`
        ]

        [
          `T temp(t); temp &= u`.

          Return convertible to `T`. See the [link utility.utilities.operators.arithmetic.smpl_oprs.symmetry Symmetry Note].
        ]

        [No]
      ]

      [
        [
           [#utility.utilities.operators.arithmetic.smpl_oprs.xorable1]`xorable<T>`

           `xorable1<T>`
        ]

        [
            `T operator^(const T&, const T&)`
        ]

        [
            `T temp(t); temp ^= t1`.

             Return convertible to `T`. See the [link utility.utilities.operators.arithmetic.smpl_oprs.symmetry Symmetry Note].
        ]

        [No]
      ]

      [
         [
             [#utility.utilities.operators.arithmetic.smpl_oprs.xorable2]`xorable<T, U>`

             `xorable2<T, U>`
         ]

         [
            `T operator^(const T&, const U&)`

            `T operator^(const U&, const T&)`
         ]

         [
             `T temp(t); temp ^= u`.

             Return convertible to `T`. See the [link utility.utilities.operators.arithmetic.smpl_oprs.symmetry Symmetry Note].
         ]

         [No]
      ]

      [
        [
           [#utility.utilities.operators.arithmetic.smpl_oprs.incrementable]`incrementable<T>`
        ]

        [
           `T operator++(T&, int)`
        ]

        [
           `T temp(t); ++t`

           Return convertible to `T`.
        ]

        [No]
      ]

      [
        [
            [#utility.utilities.operators.arithmetic.smpl_oprs.decrementable]`decrementable<T>`
        ]

        [
            `T operator--(T&, int)`
        ]

        [
          `T temp(t); --t;`

           Return convertible to `T`.
        ]

        [No]
      ]

      [
        [
            [#utility.utilities.operators.arithmetic.smpl_oprs.left_shiftable1]`left_shiftable<T>`

            `left_shiftable1<T>`
        ]

        [
            `T operator<<(const T&, const T&)`
        ]

        [
            `T temp(t); temp <<= t1`.

            Return convertible to `T`. See the [link utility.utilities.operators.arithmetic.smpl_oprs.symmetry Symmetry Note].
        ]

        [No]
      ]

      [
        [
            [#utility.utilities.operators.arithmetic.smpl_oprs.left_shiftable2]`left_shiftable<T,U>`

            `left_shiftable2<T, U>`
        ]

        [
            `T operator<<(const T&, const U&)`
        ]

        [
            `T temp(t); temp <<= u`.

            Return convertible to `T`. See the [link utility.utilities.operators.arithmetic.smpl_oprs.symmetry Symmetry Note].
        ]

        [No]
      ]

      [
         [
             [#utility.utilities.operators.arithmetic.smpl_oprs.right_shiftable1]`right_shiftable<T>`

             `right_shiftable1<T>`
         ]

         [
             `T operator>>(const T&, const T&)`
         ]

         [
             `T temp(t); temp >>= t1`.

             Return convertible to `T`. See the [link utility.utilities.operators.arithmetic.smpl_oprs.symmetry Symmetry Note].
         ]

         [No]
      ]

      [
        [
            [#utility.utilities.operators.arithmetic.smpl_oprs.right_shiftable2]`right_shiftable<T,U>`

            `right_shiftable2<T, U>`
        ]

        [
            `T operator>>(const T&, const U&)`
        ]

        [
            `T temp(t); temp >>= u`.

            Return convertible to `T`. See the [link utility.utilities.operators.arithmetic.smpl_oprs.symmetry Symmetry Note].
        ]

        [No]
      ]

      [
        [
            [#utility.utilities.operators.arithmetic.smpl_oprs.equivalent1]`equivalent<T>`

            __equivalent1__`<T>`
        ]

        [
            `bool operator==(const T&, const T&)`
        ]

        [
            `t < t1`.

            Return convertible to `bool`. See the [link utility.utilities.operators.arithmetic.smpl_oprs.ordering Ordering Note].
        ]

        [
           Since `C++11`, except [@https://developercommunity.visualstudio.com/content/problem/414193/rejects-valid-constexpr-marked-friend-function-def.html MSVC < v19.22]
        ]
      ]

      [
        [
            [#utility.utilities.operators.arithmetic.smpl_oprs.equivalent2]`equivalent<T, U>`

            __equivalent2__`<T, U>`
        ]

        [
            `bool operator==(const T&, const U&)`
        ]

        [
            `t < u`. `t > u`.

             Returns convertible to `bool`. See the [link utility.utilities.operators.arithmetic.smpl_oprs.ordering Ordering Note].
        ]

        [
           Since `C++11`, except [@https://developercommunity.visualstudio.com/content/problem/414193/rejects-valid-constexpr-marked-friend-function-def.html MSVC < v19.22]
        ]
      ]

      [
        [
            [#utility.utilities.operators.arithmetic.smpl_oprs.partially_ordered1]`partially_ordered<T>`

             __partially_ordered1__`<T>`
        ]

        [
         `bool operator>(const T&, const T&)`

         `bool operator<=(const T&, const T&)`

         `bool operator>=(const T&, const T&)`
        ]

        [
           `t < t1`. `t == t1`.

            Returns convertible to `bool`. See the [link utility.utilities.operators.arithmetic.smpl_oprs.ordering Ordering Note].
        ]

        [
           Since `C++11`, except [@https://developercommunity.visualstudio.com/content/problem/414193/rejects-valid-constexpr-marked-friend-function-def.html MSVC < v19.22]
        ]
      ]

      [
        [
          [#utility.utilities.operators.arithmetic.smpl_oprs.partially_ordered2]`partially_ordered<T,U>`

         __partially_ordered2__`<T, U>`
        ]

        [
         `bool operator<=(const T&, const U&)`

         `bool operator>=(const T&, const U&)`

         `bool operator>(const U&, const T&)`

         `bool operator<(const U&, const T&)`

         `bool operator<=(const U&, const T&)`

         `bool operator>=(const U&, const T&)`
        ]

        [
            `t < u`. `t > u`. `t == u`.

            Returns convertible to `bool`. See the [link utility.utilities.operators.arithmetic.smpl_oprs.ordering Ordering Note].
        ]

        [
           Since `C++11`, except [@https://developercommunity.visualstudio.com/content/problem/414193/rejects-valid-constexpr-marked-friend-function-def.html MSVC < v19.22]
        ]
      ]
]

[#utility.utilities.operators.arithmetic.smpl_oprs.ordering]
['Ordering Note]: The [link utility.utilities.operators.arithmetic.smpl_oprs.less_than_comparable1 `less_than_comparable<T>`] and
[link utility.utilities.operators.arithmetic.smpl_oprs.partially_ordered1 `partially_ordered<T>`] templates provide the same set
of operations. However, the workings of [link utility.utilities.operators.arithmetic.smpl_oprs.less_than_comparable1 `less_than_comparable<T>`] assume
that all values of type `T` can be placed in a total order. If
that is not true (e.g. Not-a-Number values in IEEE floating point
arithmetic), then [link utility.utilities.operators.arithmetic.smpl_oprs.partially_ordered1 `partially_ordered<T>`] should be
used. The [link utility.utilities.operators.arithmetic.smpl_oprs.partially_ordered1 `partially_ordered<T>`] template can
be used for a totally-ordered type, but it is not as efficient as
[link utility.utilities.operators.arithmetic.smpl_oprs.less_than_comparable1 `less_than_comparable<T>`]. This
rule also applies for [link utility.utilities.operators.arithmetic.smpl_oprs.less_than_comparable2 `less_than_comparable<T, U>`] and
[link utility.utilities.operators.arithmetic.smpl_oprs.partially_ordered2 `partially_ordered<T,U>`] with respect to the ordering
of all `T` and `U` values, and for both versions of
[link utility.utilities.operators.arithmetic.smpl_oprs.equivalent1 `equivalent<>`]. The solution for
[link utility.utilities.operators.arithmetic.smpl_oprs.equivalent1 `equivalent<>`] is to write a custom
`operator==` for the target class.

[#utility.utilities.operators.arithmetic.smpl_oprs.symmetry]
['Symmetry Note]: Before talking about symmetry, we need to talk about optimizations to
understand the reasons for the different implementation styles of
operators. Let's have a look at `operator+` for a class
`T` as an example:

```
T operator+( const T& lhs, const T& rhs )
{
    return T( lhs ) += rhs;
}
```

This would be a normal implementation of `operator+`, but it
is not an efficient one. An unnamed local copy of `lhs` is
created, `operator+=` is called on it and it is copied to the
function return value (which is another unnamed object of type
`T`). The standard doesn't generally allow the intermediate
object to be optimized away:

[:['C++11 [sect]3.7.3/3 \[basic.stc.auto\]: Automatic storage duration:] If a variable with automatic storage
duration has initialization or a destructor with side effects, it shall not be destroyed before the end of
its block, nor shall it be eliminated as an optimization even if it appears to be unused, except that a class
object or its copy/move may be eliminated as specified in 12.8.]

The reference to [sect]12.8 is important for us:

[:['C++11 [sect]12.8/31 \[class.copy\]: Copying and moving class objects:] When certain criteria are met, an implementation is allowed to omit the copy/move construction of a class
object, even if the copy/move constructor and/or destructor for the object have side effects. ([hellip])  This elision of copy/move
operations, called ['copy elision], is permitted in the following circumstances (which may be combined to
eliminate multiple copies):

[mdash] in a `return` statement in a function with a class return type, when the expression is the name of a
non-volatile automatic object (other than a function or catch-clause parameter) with the same cv-
unqualified type as the function return type, the copy/move operation can be omitted by constructing
the automatic object directly into the function's return value

([hellip])]

This optimization is known as the named return value optimization ([@https://en.cppreference.com/w/cpp/language/copy_elision#Non-mandatory_elision_of_copy.2Fmove_.28since_C.2B.2B11.29_operations NRVO]),
which leads us to the following implementation for `operator+`:

```
T operator+( const T& lhs, const T& rhs )
{
    T nrv( lhs );
    nrv += rhs;
    return nrv;
}
```

Given this implementation, the compiler is allowed to remove the
intermediate object. Sadly, not all compilers implement the NRVO, some
even implement it in an incorrect way which makes it useless here.
Without the NRVO, the NRVO-friendly code is no worse than the original
code showed above, but there is another possible implementation, which
has some very special properties:

```
T operator+( T lhs, const T& rhs )
{
    return lhs += rhs;
}
```

The difference to the first implementation is that `lhs` is
not taken as a constant reference used to create a copy; instead,
`lhs` is a by-value parameter, thus it is already the copy
needed. This allows another optimization (C++11 [sect]12.2/2 \[class.temporary\])
for some cases.

Consider `a + b + c` where the result of `a + b` is not copied when
used as `lhs` when adding `c`. This is more efficient than the original
code, but not as efficient as a compiler using the NRVO. For most people,
it is still preferable for compilers that don't implement the NRVO, but
the `operator+` now has a different function signature. Also,
the number of objects created differs for `(a + b) + c` and `a + (b + c)`.

Most probably, this won't be a problem for you, but if your code relies on the function
signature or a strict symmetric behaviour, you should set
`BOOST_FORCE_SYMMETRIC_OPERATORS` in your user-config. This
will force the NRVO-friendly implementation to be used even for compilers
that do not implement the NRVO.

[endsect]

[section:grpd_oprs Grouped Arithmetic Operators]

The following templates provide common groups of related operations.
For example, since a type which is addable is usually also subtractable,
the [link utility.utilities.operators.arithmetic.grpd_oprs.additive1 `additive`] template provides the
combined operators of both. The grouped operator templates have an
additional optional template parameter `B`, which is not
shown, for the [link utility.utilities.operators.usage.chaining base class chaining] technique.

[table Notation
    [[Key] [Description]]
    [[`T`] [primary operand type]]
    [[`U`] [alternate operand type]]
]

[table Grouped Arithmetic Operator Template Classes
  [
    [Template]

    [Component Operator Templates]
  ]
  [
    [
        [#utility.utilities.operators.arithmetic.grpd_oprs.totally_ordered1] `totally_ordered<T>`

        __totally_ordered1__`<T>`
    ]
    [
        [link utility.utilities.operators.arithmetic.smpl_oprs.less_than_comparable1 `less_than_comparable<T>`]

        [link utility.utilities.operators.arithmetic.smpl_oprs.equality_comparable1 `equality_comparable<T>`]
    ]
  ]
  [
    [
        [#utility.utilities.operators.arithmetic.grpd_oprs.totally_ordered2]`totally_ordered<T,U>`

        __totally_ordered2__`<T, U>`
     ]
     [
        [link utility.utilities.operators.arithmetic.smpl_oprs.less_than_comparable2 `less_than_comparable<T,U>`]

        [link utility.utilities.operators.arithmetic.smpl_oprs.equality_comparable2 `equality_comparable<T,U>`]
     ]
  ]
  [
    [
       [#utility.utilities.operators.arithmetic.grpd_oprs.additive1]`additive<T>`

       __additive1__`<T>`
     ]
     [
        [link utility.utilities.operators.arithmetic.smpl_oprs.addable1 `addable<T>`]

        [link utility.utilities.operators.arithmetic.smpl_oprs.subtractable1 `subtractable<T>`]
     ]
  ]
  [
    [
       [#utility.utilities.operators.arithmetic.grpd_oprs.additive2]`additive<T, U>`

       __additive2__`<T, U>`
     ]
    [
        [link utility.utilities.operators.arithmetic.smpl_oprs.addable2 `addable<T, U>`]

        [link utility.utilities.operators.arithmetic.smpl_oprs.subtractable2 `subtractable<T,U>`]
    ]
  ]
  [
    [
        [#utility.utilities.operators.arithmetic.grpd_oprs.multiplicative1] `multiplicative<T>`

        __multiplicative1__`<T>`
    ]
    [
        [link utility.utilities.operators.arithmetic.smpl_oprs.multipliable1 `multipliable<T>`]

        [link utility.utilities.operators.arithmetic.smpl_oprs.dividable1 `dividable<T>`]
    ]
  ]
  [
    [
       [#utility.utilities.operators.arithmetic.grpd_oprs.multiplicative2]`multiplicative<T,U>`

       __multiplicative2__`<T, U>`
    ]
    [
        [link utility.utilities.operators.arithmetic.smpl_oprs.multipliable2 `multipliable<T,U>`]

        [link utility.utilities.operators.arithmetic.smpl_oprs.dividable2 `dividable<T,U>`]
    ]
  ]
  [
     [
        [#utility.utilities.operators.arithmetic.grpd_oprs.integer_multiplicative1] `integer_multiplicative<T>`

        __integer_multiplicative1__`<T>`
     ]
     [
        [link utility.utilities.operators.arithmetic.grpd_oprs.multiplicative1 `multiplicative<T>`]

        [link utility.utilities.operators.arithmetic.smpl_oprs.modable1 `modable<T>`]
     ]
  ]
  [
     [
        [#utility.utilities.operators.arithmetic.grpd_oprs.integer_multiplicative2] `integer_multiplicative<T,U>`

        __integer_multiplicative2__`<T, U>`
     ]
     [
         [link utility.utilities.operators.arithmetic.grpd_oprs.multiplicative2 `multiplicative<T,U>`]

         [link utility.utilities.operators.arithmetic.smpl_oprs.modable2 `modable<T, U>`]
     ]
  ]
  [
    [
        [#utility.utilities.operators.arithmetic.grpd_oprs.arithmetic1]`arithmetic<T>`

        __arithmetic1__`<T>`
     ]
     [
        [link utility.utilities.operators.arithmetic.grpd_oprs.additive1 `additive<T>`]

        [link utility.utilities.operators.arithmetic.grpd_oprs.multiplicative1 `multiplicative<T>`]
     ]
  ]
  [
    [
        [#utility.utilities.operators.arithmetic.grpd_oprs.arithmetic2]`arithmetic<T, U>`

        __arithmetic2__`<T, U>`
    ]
    [
        [link utility.utilities.operators.arithmetic.grpd_oprs.additive2 `additive<T,U>`]

        [link utility.utilities.operators.arithmetic.grpd_oprs.multiplicative2 `multiplicative<T,U>`]
    ]
  ]
  [
    [
       [#utility.utilities.operators.arithmetic.grpd_oprs.integer_arithmetic1] `integer_arithmetic<T>`

       __integer_arithmetic1__`<T>`
    ]
    [
        [link utility.utilities.operators.arithmetic.grpd_oprs.additive1 `additive<T>`]

        [link utility.utilities.operators.arithmetic.grpd_oprs.integer_multiplicative1 `integer_multiplicative<T>`]
    ]
  ]
  [
    [
        [#utility.utilities.operators.arithmetic.grpd_oprs.integer_arithmetic2]`integer_arithmetic<T, U>`

        __integer_arithmetic2__`<T, U>`
    ]
    [
        [link utility.utilities.operators.arithmetic.grpd_oprs.additive2 `additive<T,U>`]

        [link utility.utilities.operators.arithmetic.grpd_oprs.integer_multiplicative2 `integer_multiplicative<T,U>`]
    ]
  ]
  [
    [
       [#utility.utilities.operators.arithmetic.grpd_oprs.bitwise1]`bitwise<T>`

       __bitwise1__`<T>`
    ]
    [
        [link utility.utilities.operators.arithmetic.smpl_oprs.xorable1 `xorable<T>`]

        [link utility.utilities.operators.arithmetic.smpl_oprs.andable1 `andable<T>`]

        [link utility.utilities.operators.arithmetic.smpl_oprs.orable1 `orable<T>`]
    ]
  ]
  [
    [
        [#utility.utilities.operators.arithmetic.grpd_oprs.bitwise2]`bitwise<T, U>`

        __bitwise2__`<T, U>`
    ]
    [
        [link utility.utilities.operators.arithmetic.smpl_oprs.xorable2 `xorable<T, U>`]

        [link utility.utilities.operators.arithmetic.smpl_oprs.andable2 `andable<T, U>`]

        [link utility.utilities.operators.arithmetic.smpl_oprs.orable2 `orable<T, U>`]
    ]
  ]
  [
    [
        [#utility.utilities.operators.arithmetic.grpd_oprs.unit_steppable] `unit_steppable<T>`
    ]
    [
        [link utility.utilities.operators.arithmetic.smpl_oprs.incrementable `incrementable<T>`]

        [link utility.utilities.operators.arithmetic.smpl_oprs.decrementable `decrementable<T>`]
    ]
  ]
  [
    [
        [#utility.utilities.operators.arithmetic.grpd_oprs.shiftable1]`shiftable<T>`

        __shiftable1__`<T>`
    ]
    [
        [link utility.utilities.operators.arithmetic.smpl_oprs.left_shiftable1 `left_shiftable<T>`]

        [link utility.utilities.operators.arithmetic.smpl_oprs.right_shiftable1 `right_shiftable<T>`]
    ]
  ]
  [
    [
        [#utility.utilities.operators.arithmetic.grpd_oprs.shiftable2]`shiftable<T, U>`

        __shiftable2__`<T, U>`
    ]
    [
        [link utility.utilities.operators.arithmetic.smpl_oprs.left_shiftable2 `left_shiftable<T,U>`]

        [link utility.utilities.operators.arithmetic.smpl_oprs.right_shiftable2 `right_shiftable<T,U>`]
    ]
  ]
  [
     [
         [#utility.utilities.operators.arithmetic.grpd_oprs.ring_operators1] `ring_operators<T>`

         __ring_operators1__`<T>`
     ]
     [
        [link utility.utilities.operators.arithmetic.grpd_oprs.additive1 `additive<T>`]

        [link utility.utilities.operators.arithmetic.smpl_oprs.multipliable1 `multipliable<T>`]
    ]
  ]
  [
    [
       [#utility.utilities.operators.arithmetic.grpd_oprs.ring_operators2]`ring_operators<T,U>`

       __ring_operators2__`<T, U>`
    ]
    [
        [link utility.utilities.operators.arithmetic.grpd_oprs.additive2 `additive<T,U>`]

        [link utility.utilities.operators.arithmetic.smpl_oprs.subtractable2_left `subtractable2_left<T,U>`]

        [link utility.utilities.operators.arithmetic.smpl_oprs.multipliable2 `multipliable<T,U>`]
    ]
  ]
  [
    [
        [#utility.utilities.operators.arithmetic.grpd_oprs.ordered_ring_operators1] `ordered_ring_operators<T>`

        __ordered_ring_operators1__`<T>`
    ]
    [
        [link utility.utilities.operators.arithmetic.grpd_oprs.ring_operators1 `ring_operators<T>`]

        [link utility.utilities.operators.arithmetic.grpd_oprs.totally_ordered1 `totally_ordered<T>`]
    ]
  ]
  [
     [
        [#utility.utilities.operators.arithmetic.grpd_oprs.ordered_ring_operators2] `ordered_ring_operators<T,U>`

        __ordered_ring_operators2__`<T, U>`
     ]
     [
        [link utility.utilities.operators.arithmetic.grpd_oprs.ring_operators2 `ring_operators<T,U>`]

        [link utility.utilities.operators.arithmetic.grpd_oprs.totally_ordered2 `totally_ordered<T,U>`]
     ]
  ]
  [
    [
        [#utility.utilities.operators.arithmetic.grpd_oprs.field_operators1]`field_operators<T>`

        __field_operators1__`<T>`
    ]
    [
        [link utility.utilities.operators.arithmetic.grpd_oprs.ring_operators1 `ring_operators<T>`]

        [link utility.utilities.operators.arithmetic.smpl_oprs.dividable1 `dividable<T>`]
    ]
  ]
  [
    [
       [#utility.utilities.operators.arithmetic.grpd_oprs.field_operators2]`field_operators<T,U>`

       __field_operators2__`<T, U>`
    ]
    [
        [link utility.utilities.operators.arithmetic.grpd_oprs.ring_operators2 `ring_operators<T,U>`]

        [link utility.utilities.operators.arithmetic.smpl_oprs.dividable2 `dividable<T,U>`]

        [link utility.utilities.operators.arithmetic.smpl_oprs.dividable2_left `dividable2_left<T,U>`]
    ]
  ]
  [
    [
        [#utility.utilities.operators.arithmetic.grpd_oprs.ordered_field_operators1]`ordered_field_operators<T>`

        __ordered_field_operators1__`<T>`
    ]
    [
        [link utility.utilities.operators.arithmetic.grpd_oprs.field_operators1 `field_operators<T>`]

        [link utility.utilities.operators.arithmetic.grpd_oprs.totally_ordered1 `totally_ordered<T>`]
    ]
  ]
  [
    [
         [#utility.utilities.operators.arithmetic.grpd_oprs.ordered_field_operators2]`ordered_field_operators<T,U>`

         __ordered_field_operators2__`<T, U>`
    ]
    [
        [link utility.utilities.operators.arithmetic.grpd_oprs.field_operators2 `field_operators<T,U>`]

        [link utility.utilities.operators.arithmetic.grpd_oprs.totally_ordered2 `totally_ordered<T,U>`]
    ]
  ]
  [
    [
        [#utility.utilities.operators.arithmetic.grpd_oprs.euclidean_ring_operators1]`euclidean_ring_operators<T>`

        __euclidean_ring_operators1__`<T>`
    ]
    [
        [link utility.utilities.operators.arithmetic.grpd_oprs.ring_operators1 `ring_operators<T>`]

        [link utility.utilities.operators.arithmetic.smpl_oprs.dividable1 `dividable<T>`]

        [link utility.utilities.operators.arithmetic.smpl_oprs.modable1 `modable<T>`]
    ]
  ]
  [
    [
        [#utility.utilities.operators.arithmetic.grpd_oprs.euclidean_ring_operators2] `euclidean_ring_operators<T,U>`

        __euclidean_ring_operators2__`<T, U>`
    ]
    [
        [link utility.utilities.operators.arithmetic.grpd_oprs.ring_operators2 `ring_operators<T,U>`]

        [link utility.utilities.operators.arithmetic.smpl_oprs.dividable2 `dividable<T,U>`]

        [link utility.utilities.operators.arithmetic.smpl_oprs.dividable2_left `dividable2_left<T,U>`]

        [link utility.utilities.operators.arithmetic.smpl_oprs.modable2 `modable<T, U>`]

        [link utility.utilities.operators.arithmetic.smpl_oprs.modable2_left `modable2_left<T,U>`]
    ]
  ]
  [
    [
         [#utility.utilities.operators.arithmetic.grpd_oprs.ordered_euclidean_ring_operators1] `ordered_euclidean_ring_operators<T>`

         __ordered_euclidean_ring_operators1__`<T>`
    ]
    [
        [link utility.utilities.operators.arithmetic.grpd_oprs.euclidean_ring_operators1 `euclidean_ring_operators<T>`]

        [link utility.utilities.operators.arithmetic.grpd_oprs.totally_ordered1 `totally_ordered<T>`]
    ]
  ]
  [
     [
         [#utility.utilities.operators.arithmetic.grpd_oprs.ordered_euclidean_ring_operators2]`ordered_euclidean_ring_operators<T,U>`

         __ordered_euclidean_ring_operators2__`<T, U>`
     ]
     [
        [link utility.utilities.operators.arithmetic.grpd_oprs.euclidean_ring_operators2 `euclidean_ring_operators<T,U>`]

        [link utility.utilities.operators.arithmetic.grpd_oprs.totally_ordered2 `totally_ordered<T,U>`]
     ]
  ]
]

['Spelling: euclidean vs. euclidian]: Older versions of the
Boost.Operators library used "`euclidian`", but it was pointed out that
"`euclidean`" is the more common spelling. To be compatible with older
version, the library now supports both spellings.

[endsect]

[section:ex_oprs Example Templates]

The arithmetic operator class templates [link utility.utilities.operators.arithmetic.ex_oprs.operators1 `operators<>`] and
[link utility.utilities.operators.arithmetic.ex_oprs.operators2 `operators2<>`] are examples of non-extensible operator
grouping classes. These legacy class templates, from previous versions
of the header, cannot be used for [link utility.utilities.operators.usage.chaining base class chaining].

[table Notation
    [[Key] [Description]]
    [[`T`] [primary operand type]]
    [[`U`] [alternate operand type]]
]

[table Final Arithmetic Operator Template Classes
  [
    [Template]

    [Component Operator Templates]
  ]
  [
    [
       [#utility.utilities.operators.arithmetic.ex_oprs.operators1]__operators__`<T>`
    ]
    [
        [link utility.utilities.operators.arithmetic.grpd_oprs.totally_ordered1 `totally_ordered<T>`]

        [link utility.utilities.operators.arithmetic.grpd_oprs.integer_arithmetic1 `integer_arithmetic<T>`]

        [link utility.utilities.operators.arithmetic.grpd_oprs.bitwise1 `bitwise<T>`]

        [link utility.utilities.operators.arithmetic.grpd_oprs.unit_steppable `unit_steppable<T>`]
    ]
  ]
  [
    [
        [#utility.utilities.operators.arithmetic.ex_oprs.operators2]__operators__`<T, U>`

        __operators2__`<T, U>`
    ]
    [
        [link utility.utilities.operators.arithmetic.grpd_oprs.totally_ordered2 `totally_ordered<T,U>`]

        [link utility.utilities.operators.arithmetic.grpd_oprs.integer_arithmetic2 `integer_arithmetic<T,U>`]

        [link utility.utilities.operators.arithmetic.grpd_oprs.bitwise2 `bitwise<T, U>`]
    ]
  ]
]

[#utility.utilities.operators.arithmetic.grpd_oprs.a_demo]
['Arithmetic Operators Demonstration and Test Program]: The
[@../../../test/operators_test.cpp `operators_test.cpp`]
program demonstrates the use of the arithmetic operator templates, and
can also be used to verify correct operation. Check the compiler status
report for the test results with selected platforms.

[endsect]

[endsect]

[section:deref Dereference Operators and Iterator Helpers]

The [link utility.utilities.operators.deref.iterator iterator helper] templates ease the task of
creating a custom iterator. Similar to arithmetic types, a complete
iterator has many operators that are "redundant" and can be implemented
in terms of the core set of operators.

The [link utility.utilities.operators.deref.dereference dereference operators] were motivated by
the [link utility.utilities.operators.deref.iterator iterator helpers], but are often useful in
non-iterator contexts as well. Many of the redundant iterator operators
are also arithmetic operators, so the iterator helper classes borrow many
of the operators defined above. In fact, only two new operators need to
be defined: the pointer-to-member `operator->` and the
subscript `operator[]`.

The requirements for the types used to instantiate the dereference
operators are specified in terms of expressions which must be valid and
their return type. The composite operator templates list their component
templates, which the instantiating type must support, and possibly other
requirements.

[section:dereference Dereference Operators]

All the dereference operator templates in this table accept an
optional template parameter (not shown) to be used for
[link utility.utilities.operators.usage.chaining base class chaining].

[table Notation
    [[Key] [Description]]
    [[`T`] [operand type]]
    [[`D`] [`difference_type`]]
    [[`i`] [object of type `T` (an iterator)]]
    [[`P`] [`pointer` type]]
    [[`R`] [`reference` type]]
    [[`n`] [object of type `D` (an index)]]
]

[table Dereference Operator Template Classes
  [
      [Template]

      [Supplied Operations]

      [Requirements]
  ]
  [
    [
      [#utility.utilities.operators.deref.dereference.dereferenceable]__dereferenceable__`<T,P>`
    ]
    [
      `P operator->() const`
    ]
    [
      `*i`. Address of the returned value convertible to `P`.
    ]
  ]
  [
    [
       [#utility.utilities.operators.deref.dereference.indexable]__indexable__`<T, D, R>`
    ]
    [
       `R operator[](D n) const`
    ]
    [
       `*(i + n)`. Return of type `R`.
    ]
  ]
]

[endsect]

[section:grpd_iter_oprs Grouped Iterator Operators]

There are five iterator operator class templates, each for a different
category of iterator. The following table shows the operator groups for
any category that a custom iterator could define. These class templates
have an additional optional template parameter `B`, which is
not shown, to support [link utility.utilities.operators.usage.chaining base class chaining].

[table Notation
    [[Key] [Description]]
    [[`T`] [operand type]]
    [[`D`] [`difference_type`]]
    [[`V`] [`value_type`]]
    [[`P`] [`pointer` type]]
    [[`R`] [`reference` type]]
]

[table Iterator Operator Class Templates
  [
    [Template]

    [Component Operator Templates]
  ]
  [
    [
        [#utility.utilities.operators.deref.grpd_iter_oprs.input_iteratable]__input_iteratable__`<T, P>`
    ]
    [
        [link utility.utilities.operators.arithmetic.smpl_oprs.equality_comparable1 `equality_comparable<T>`]

        [link utility.utilities.operators.arithmetic.smpl_oprs.incrementable `incrementable<T>`]

        [link utility.utilities.operators.deref.dereference.dereferenceable `dereferenceable<T,P>`]
    ]
  ]
  [
    [
        [#utility.utilities.operators.deref.grpd_iter_oprs.output_iteratable]__output_iteratable__`<T>`
    ]
    [
        [link utility.utilities.operators.arithmetic.smpl_oprs.incrementable `incrementable<T>`]
    ]
  ]
  [
    [
        [#utility.utilities.operators.deref.grpd_iter_oprs.forward_iteratable]__forward_iteratable__`<T, P>`
    ]
    [
        [link utility.utilities.operators.deref.grpd_iter_oprs.input_iteratable `input_iteratable<T,P>`]
    ]
  ]
  [
    [
        [#utility.utilities.operators.deref.grpd_iter_oprs.bidirectional_iteratable] __bidirectional_iteratable__`<T,P>`
    ]
    [
        [link utility.utilities.operators.deref.grpd_iter_oprs.forward_iteratable `forward_iteratable<T, P>`]

        [link utility.utilities.operators.arithmetic.smpl_oprs.decrementable `decrementable<T>`]
    ]
  ]
  [
    [
        [#utility.utilities.operators.deref.grpd_iter_oprs.random_access_iteratable] __random_access_iteratable__`<T, P, D, R>`
    ]
    [
        [link utility.utilities.operators.deref.grpd_iter_oprs.bidirectional_iteratable `bidirectional_iteratable<T, P>`]

        [link utility.utilities.operators.arithmetic.grpd_oprs.totally_ordered1 `totally_ordered<T>`]

        [link utility.utilities.operators.arithmetic.grpd_oprs.additive2 `additive<T, D>`]

        [link utility.utilities.operators.deref.dereference.indexable `indexable<T, D, R>`]
    ]
  ]
]

[endsect]

[section:iterator Iterator Helpers]

There are also five iterator helper class templates, each
corresponding to a different iterator category. These classes cannot be
used for [link utility.utilities.operators.usage.chaining base class chaining]. The following
summaries show that these class templates supply both the iterator
operators from the [link utility.utilities.operators.deref.grpd_iter_oprs iterator operator class
templates] and the iterator `typedef`s required by the C++ standard,
such as `iterator_category` and `value_type`.

[table Notation
    [[Key] [Description]]
    [[`T`] [operand type]]
    [[`D`] [`difference_type`]]
    [[`V`] [`value_type`]]
    [[`P`] [`pointer` type]]
    [[`R`] [`reference` type]]
    [[`x1`, `x2`] [objects of type `T`]]
]

[table Helper Class Templates
  [
    [Template]

    [Operations and Requirements]
  ]
  [
    [
       [#utility.utilities.operators.deref.iterator.input_iterator_helper]__input_iterator_helper__`<T, V, D, P, R>`
    ]
    [
        Supports the operations and has the requirements of [link utility.utilities.operators.deref.grpd_iter_oprs.input_iteratable `input_iteratable<T, P>`]
    ]
  ]
  [
    [
        [#utility.utilities.operators.deref.iterator.output_iterator_helper]__output_iterator_helper__`<T>`
    ]
    [
        Supports the operations and has the requirements of [link utility.utilities.operators.deref.grpd_iter_oprs.output_iteratable `output_iteratable<T>`]

        See also \[[link note:1 1]\], \[[link note:2 2]\].
    ]
  ]
  [
    [
        [#utility.utilities.operators.deref.iterator.forward_iterator_helper] __forward_iterator_helper__`<T, V, D, P, R>`
    ]
    [
        Supports the operations and has the requirements of [link utility.utilities.operators.deref.grpd_iter_oprs.forward_iteratable `forward_iteratable<T, P>`]
    ]
  ]
  [
    [
        [#utility.utilities.operators.deref.iterator.bidirectional_iterator_helper] __bidirectional_iterator_helper__`<T, V, D, P, R>`
    ]
    [
        Supports the operations and has the requirements of [link utility.utilities.operators.deref.grpd_iter_oprs.bidirectional_iteratable `bidirectional_iteratable<T, P>`]
    ]
  ]
  [
    [
        [#utility.utilities.operators.deref.iterator.random_access_iterator_helper]__random_access_iterator_helper__`<T, V, D, P, R>`
    ]
    [
        Supports the operations and has the requirements of [link utility.utilities.operators.deref.grpd_iter_oprs.random_access_iteratable `random_access_iteratable<T, P, D, R>`]

        To satisfy __RandomAccessIterator__, `x1 - x2` with return convertible to `D` is also required.
    ]
  ]
]

[#utility.utilities.operators.deref.iterator.iterator_helpers_notes]
['Iterator Helper Notes]:

# [#note:1] Unlike other iterator helpers templates, __output_iterator_helper__ takes only one template parameter -
the type of its target class. Although to some it might seem like an unnecessary restriction, the standard requires
`difference_type` and `value_type` of any output iterator to be `void` (C++11 [sect]24.4.1 \[lib.iterator.traits\]), and
__output_iterator_helper__ template respects this requirement. Also, output iterators in the standard have void `pointer` and
`reference` types, so the __output_iterator_helper__ does the same.
# [#note:2] As self-proxying is the easiest and most common way to implement output iterators (see,
for example, insert (C++11 [sect]24.5.2 \[insert.iterators\]) and stream iterators (C++11 [sect]24.6 \[stream.iterators\]) in the standard library),
__output_iterator_helper__ supports the idiom by defining `operator*` and `operator++` member functions which
just return a non-const reference to the iterator itself. Support for self-proxying allows us, in many cases,
to reduce the task of writing an output iterator to writing just two member functions - an appropriate
constructor and a copy-assignment operator. For example, here is a possible implementation of
[@boost:/libs/iterator/doc/function_output_iterator.html `boost::function_output_iterator`] adaptor:

```
template<class UnaryFunction>
struct function_output_iterator
    : boost::__output_iterator_helper__< function_output_iterator<UnaryFunction> >
{
    explicit function_output_iterator(UnaryFunction const& f = UnaryFunction())
        : func(f) {}

    template<typename T>
    function_output_iterator& operator=(T const& value)
    {
        this->func(value);
        return *this;
    }

private:
    UnaryFunction func;
};
```

Note that support for self-proxying does not prevent you from using
__output_iterator_helper__ to ease any other, different kind of
output iterator's implementation. If
__output_iterator_helper__'s target type provides its own
definition of `operator*` or/and `operator++`, then
these operators will get used and the ones supplied by
__output_iterator_helper__ will never be instantiated.

[endsect]

[section:i_demo Iterator Demonstration and Test Program]

The [@../../../test/iterators_test.cpp `iterators_test.cpp`]
program demonstrates the use of the iterator templates, and can also be
used to verify correct operation. The following is the custom iterator
defined in the test program. It demonstrates a correct (though trivial)
implementation of the core operations that must be defined in order for
the iterator helpers to "fill in" the rest of the iterator
operations.

```
template <class T, class R, class P>
struct test_iter
    : public boost::__random_access_iterator_helper__<
                test_iter<T, R, P>, T, __std_ptrdiff_t__, P, R
             >
{
    typedef test_iter self;
    typedef R Reference;
    typedef __std_ptrdiff_t__ Distance;

public:
    explicit test_iter(T* i =0);
    test_iter(const self& x);
    self& operator=(const self& x);
    Reference operator*() const;
    self& operator++();
    self& operator--();
    self& operator+=(Distance n);
    self& operator-=(Distance n);
    bool operator==(const self& x) const;
    bool operator<(const self& x) const;
    friend Distance operator-(const self& x, const self& y);
};
```

Check the [@http://www.boost.org/development/testing.html compiler status report] for the
test results with selected platforms.

[endsect]

[endsect]

[section:old_lib_note Note for Users of Older Versions]

The [link utility.utilities.operators.usage.chaining changes in the library interface and
recommended usage] were motivated by some practical issues described
below. The new version of the library is still backward-compatible with
the former one, so you are not ['forced] to change any existing code,
but the old usage is deprecated.

Though it was arguably simpler and more intuitive than using
[link utility.utilities.operators.usage.chaining base class chaining], it has been discovered
that the old practice of deriving from multiple operator
templates can cause the resulting classes to be much larger than they
should be. Most modern C++ compilers significantly bloat the size of
classes derived from multiple empty base classes, even though the base
classes themselves have no state. For instance, the size of
`point<int>` from the [link utility.utilities.operators.example example]
above was 12-24 bytes on various compilers for the Win32 platform,
instead of the expected 8 bytes.

Strictly speaking, it was not the library's fault [ndash] the language rules
allow the compiler to apply the [@https://en.cppreference.com/w/cpp/language/ebo empty base class optimization]
in that situation. In principle an arbitrary number of empty base classes can
be allocated at the same offset, provided that none of them have a common
ancestor (see [sect]10 \[class.derived\] paragraph 8 of the C++11 standard).

But the language definition also does not ['require] implementations
to do the optimization, and few if any of today's compilers implement it
when multiple inheritance is involved. What's worse, it is very unlikely
that implementors will adopt it as a future enhancement to existing
compilers, because it would break binary compatibility between code
generated by two different versions of the same compiler. As Matt Austern
said, "One of the few times when you have the freedom to do this sort of
thing is when you are targeting a new architecture[hellip]". On the other hand,
many common compilers will use the empty base optimization for single
inheritance hierarchies.

Given the importance of the issue for the users of the library (which
aims to be useful for writing light-weight classes like
`MyInt` or `point<>`), and the forces described above, we decided to change
the library interface so that the object size bloat could be eliminated even
on compilers that support only the simplest form of the empty base class
optimization. The current library interface is the result of those changes.
Though the new usage is a bit more complicated than the old one, we think
it's worth it to make the library more useful in real world. Alexy Gurtovoy
contributed the code which supports the new usage idiom while allowing the
library to remain backward-compatible.

[endsect]

[section:contributors Acknowledgments]

* [@http://www.boost.org/people/dave_abrahams.htm Dave Abrahams]:
  Started the library and contributed the arithmetic operators in
  [@../../../../boost/operators.hpp `boost/operators.hpp`].

* [@http://www.boost.org/people/jeremy_siek.htm Jeremy Siek]:
  Contributed the [link utility.utilities.operators.deref dereference operators and iterator
  helpers] in [@../../../../boost/operators.hpp boost/operators.hpp].
  Also contributed [@../../../test/iterators_test.cpp iterators_test.cpp].

* [@http://www.boost.org/people/aleksey_gurtovoy.htm Aleksey Gurtovoy]:
  Contributed the code to support [link utility.utilities.operators.usage.chaining base class
  chaining] while remaining backward-compatible with old versions of
  the library.

* [@http://www.boost.org/people/beman_dawes.html Beman Dawes]:
  Contributed [@../../../test/operators_test.cpp `operators_test.cpp`].

* [@http://www.boost.org/people/daryle_walker.html Daryle Walker]:
  Contributed classes for the shift operators, equivalence, partial
  ordering, and arithmetic conversions. Added the grouped operator
  classes. Added helper classes for input and output iterators.

* Helmut Zeisel: Contributed the 'left' operators and added some grouped operator
  classes.

* Daniel Frey: Contributed the NRVO-friendly and symmetric implementation of
  arithmetic operators.

[endsect]

[endsect]
