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 /*!
 @file
 Forward declares `boost::hana::Group`.
 
 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_GROUP_HPP
 #define BOOST_HANA_FWD_CONCEPT_GROUP_HPP
 
 #include <boost/hana/config.hpp>
 
 
 namespace boost { namespace hana {
     //! @ingroup group-concepts
     //! @defgroup group-Group Group
     //! The `Group` concept represents `Monoid`s where all objects have
     //! an inverse w.r.t. the `Monoid`'s binary operation.
     //!
     //! A [Group][1] is an algebraic structure built on top of a `Monoid`
     //! which adds the ability to invert the action of the `Monoid`'s binary
     //! operation on any element of the set. Specifically, a `Group` is a
     //! `Monoid` `(S, +)` such that every element `s` in `S` has an inverse
     //! (say `s'`) which is such that
     //! @code
     //!     s + s' == s' + s == identity of the Monoid
     //! @endcode
     //!
     //! There are many examples of `Group`s, one of which would be the
     //! additive `Monoid` on integers, where the inverse of any integer
     //! `n` is the integer `-n`. The method names used here refer to
     //! exactly this model.
     //!
     //!
     //! Minimal complete definitions
     //! ----------------------------
     //! 1. `minus`\n
     //! When `minus` is specified, the `negate` method is defaulted by setting
     //! @code
     //!     negate(x) = minus(zero<G>(), x)
     //! @endcode
     //!
     //! 2. `negate`\n
     //! When `negate` is specified, the `minus` method is defaulted by setting
     //! @code
     //!     minus(x, y) = plus(x, negate(y))
     //! @endcode
     //!
     //!
     //! Laws
     //! ----
     //! For all objects `x` of a `Group` `G`, the following laws must be
     //! satisfied:
     //! @code
     //!     plus(x, negate(x)) == zero<G>() // right inverse
     //!     plus(negate(x), x) == zero<G>() // left inverse
     //! @endcode
     //!
     //!
     //! Refined concept
     //! ---------------
     //! `Monoid`
     //!
     //!
     //! Concrete models
     //! ---------------
     //! `hana::integral_constant`
     //!
     //!
     //! Free model for non-boolean arithmetic data types
     //! ------------------------------------------------
     //! A data type `T` is arithmetic if `std::is_arithmetic<T>::%value` is
     //! true. For a non-boolean arithmetic data type `T`, a model of `Group`
     //! is automatically defined by setting
     //! @code
     //!     minus(x, y) = (x - y)
     //!     negate(x) = -x
     //! @endcode
     //!
     //! @note
     //! The rationale for not providing a Group model for `bool` is the same
     //! as for not providing a `Monoid` model.
     //!
     //!
     //! Structure-preserving functions
     //! ------------------------------
     //! Let `A` and `B` be two `Group`s. A function `f : A -> B` is said to
     //! be a [Group morphism][2] if it preserves the group structure between
     //! `A` and `B`. Rigorously, for all objects `x, y` of data type `A`,
     //! @code
     //!     f(plus(x, y)) == plus(f(x), f(y))
     //! @endcode
     //! Because of the `Group` structure, it is easy to prove that the
     //! following will then also be satisfied:
     //! @code
     //!     f(negate(x)) == negate(f(x))
     //!     f(zero<A>()) == zero<B>()
     //! @endcode
     //! Functions with these properties interact nicely with `Group`s, which
     //! is why they are given such a special treatment.
     //!
     //!
     //! [1]: http://en.wikipedia.org/wiki/Group_(mathematics)
     //! [2]: http://en.wikipedia.org/wiki/Group_homomorphism
     template <typename G>
     struct Group;
 }} // end namespace boost::hana
 
 #endif // !BOOST_HANA_FWD_CONCEPT_GROUP_HPP

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