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 /*!
 @file
 Forward declares `boost::hana::Comparable`.
 
 Copyright Louis Dionne 2013-2022
 Distributed under the Boost Software License, Version 1.0.
 (See accompanying file LICENSE.md or copy at http://boost.org/LICENSE_1_0.txt)
  */
 
 #ifndef BOOST_HANA_FWD_CONCEPT_COMPARABLE_HPP
 #define BOOST_HANA_FWD_CONCEPT_COMPARABLE_HPP
 
 #include <boost/hana/config.hpp>
 
 
 namespace boost { namespace hana {
     //! @ingroup group-concepts
     //! @defgroup group-Comparable Comparable
     //! The `Comparable` concept defines equality and inequality.
     //!
     //! Intuitively, `Comparable` objects must define a binary predicate named
     //! `equal` that returns whether both objects represent the same abstract
     //! value. In other words, `equal` must check for deep equality. Since
     //! "representing the same abstract value" is difficult to express
     //! formally, the exact meaning of equality is partially left to
     //! interpretation by the programmer with the following guidelines:\n
     //! 1. Equality should be compatible with copy construction; copy
     //!    constructing a value yields an `equal` value.
     //! 2. Equality should be independent of representation; an object
     //!    representing a fraction as `4/8` should be `equal` to an object
     //!    representing a fraction as `2/4`, because they both represent
     //!    the mathematical object `1/2`.
     //!
     //! Moreover, `equal` must exhibit properties that make it intuitive to
     //! use for determining the equivalence of objects, which is formalized
     //! by the laws for `Comparable`.
     //!
     //!
     //! Minimal complete definition
     //! ---------------------------
     //! 1. `equal`\n
     //! When `equal` is defined, `not_equal` is implemented by default as its
     //! complement. For all objects `x`, `y` of a `Comparable` tag,
     //! @code
     //!     not_equal(x, y) == not_(equal(x, y))
     //! @endcode
     //!
     //!
     //! Laws
     //! ----
     //! `equal` must define an [equivalence relation][1], and `not_equal` must
     //! be its complement. In other words, for all objects `a`, `b`, `c` with
     //! a `Comparable` tag, the following must hold:
     //! @code
     //!     equal(a, a)                                         // Reflexivity
     //!     if equal(a, b) then equal(b, a)                     // Symmetry
     //!     if equal(a, b) && equal(b, c) then equal(a, c)      // Transitivity
     //!     not_equal(a, b) is equivalent to not_(equal(a, b))
     //! @endcode
     //!
     //!
     //! Concrete models
     //! ---------------
     //! `hana::integral_constant`, `hana::map`, `hana::optional`, `hana::pair`,
     //! `hana::range`, `hana::set`, `hana::string`, `hana::tuple`,
     //!  `hana::type`
     //!
     //!
     //! Free model for `EqualityComparable` data types
     //! ----------------------------------------------
     //! Two data types `T` and `U` that model the cross-type EqualityComparable
     //! concept presented in [N3351][2] automatically model the `Comparable`
     //! concept by setting
     //! @code
     //!     equal(x, y) = (x == y)
     //! @endcode
     //! Note that this also makes EqualityComparable types in the
     //! [usual sense][3] models of `Comparable` in the same way.
     //!
     //!
     //! Equality-preserving functions
     //! -----------------------------
     //! Let `A` and `B` be two `Comparable` tags. A function @f$f : A \to B@f$
     //! is said to be equality-preserving if it preserves the structure of the
     //! `Comparable` concept, which can be rigorously stated as follows. For
     //! all objects `x`, `y` of tag `A`,
     //! @code
     //!     if  equal(x, y)  then  equal(f(x), f(y))
     //! @endcode
     //! Equivalently, we simply require that `f` is a function in the usual
     //! mathematical sense. Another property is [injectivity][4], which can be
     //! viewed as being a "lossless" mapping. This property can be stated as
     //! @code
     //!     if  equal(f(x), f(y))  then  equal(x, y)
     //! @endcode
     //! This is equivalent to saying that `f` maps distinct elements to
     //! distinct elements, hence the "lossless" analogy. In other words, `f`
     //! will not collapse distinct elements from its domain into a single
     //! element in its image, thus losing information.
     //!
     //! These functions are very important, especially equality-preserving
     //! ones, because they allow us to reason simply about programs. Also
     //! note that the property of being equality-preserving is taken for
     //! granted in mathematics because it is part of the definition of a
     //! function. We feel it is important to make the distinction here
     //! because programming has evolved differently and as a result
     //! programmers are used to work with functions that do not preserve
     //! equality.
     //!
     //!
     //! Cross-type version of the methods
     //! ---------------------------------
     //! The `equal` and `not_equal` methods are "overloaded" to handle
     //! distinct tags with certain properties. Specifically, they are
     //! defined for _distinct_ tags `A` and `B` such that
     //! 1. `A` and `B` share a common tag `C`, as determined by the
     //!    `common` metafunction
     //! 2. `A`, `B` and `C` are all `Comparable` when taken individually
     //! 3. @f$ \mathtt{to<C>} : A \to C @f$ and @f$\mathtt{to<C>} : B \to C@f$
     //!    are both equality-preserving and injective (i.e. they are embeddings),
     //!    as determined by the `is_embedding` metafunction.
     //!
     //! The method definitions for tags satisfying the above properties are
     //! @code
     //!     equal(x, y)     = equal(to<C>(x), to<C>(y))
     //!     not_equal(x, y) = not_equal(to<C>(x), to<C>(y))
     //! @endcode
     //!
     //!
     //! Important note: special behavior of `equal`
     //! -------------------------------------------
     //! In the context of programming with heterogeneous values, it is useful
     //! to have unrelated objects compare `false` instead of triggering an
     //! error. For this reason, `equal` adopts a special behavior for
     //! unrelated objects of tags `T` and `U` that do not satisfy the above
     //! requirements for the cross-type overloads. Specifically, when `T` and
     //! `U` are unrelated (i.e. `T` can't be converted to `U` and vice-versa),
     //! comparing objects with those tags yields a compile-time false value.
     //! This has the effect that unrelated objects like `float` and
     //! `std::string` will compare false, while comparing related objects that
     //! can not be safely embedded into the same super structure (like
     //! `long long` and `float` because of the precision loss) will trigger a
     //! compile-time assertion. Also note that for any tag `T` for which the
     //! minimal complete definition of `Comparable` is not provided, a
     //! compile-time assertion will also be triggered because `T` and `T`
     //! trivially share the common tag `T`, which is the expected behavior.
     //! This design choice aims to provide more flexibility for comparing
     //! objects, while still rejecting usage patterns that are most likely
     //! programming errors.
     //!
     //!
     //! [1]: http://en.wikipedia.org/wiki/Equivalence_relation#Definition
     //! [2]: http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2012/n3351.pdf
     //! [3]: http://en.cppreference.com/w/cpp/named_req/EqualityComparable
     //! [4]: http://en.wikipedia.org/wiki/Injective_function
     template <typename T>
     struct Comparable;
 }} // end namespace boost::hana
 
 #endif // !BOOST_HANA_FWD_CONCEPT_COMPARABLE_HPP

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