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 /*!
 @file
 Forward declares `boost::hana::Applicative`.
 
 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_APPLICATIVE_HPP
 #define BOOST_HANA_FWD_CONCEPT_APPLICATIVE_HPP
 
 #include <boost/hana/config.hpp>
 
 
 namespace boost { namespace hana {
     //! @ingroup group-concepts
     //! @defgroup group-Applicative Applicative
     //! The `Applicative` concept represents `Functor`s with the ability
     //! to lift values and combine computations.
     //!
     //! A `Functor` can only take a normal function and map it over a
     //! structure containing values to obtain a new structure containing
     //! values. Intuitively, an `Applicative` can also take a value and
     //! lift it into the structure. In addition, an `Applicative` can take
     //! a structure containing functions and apply it to a structure
     //! containing values to obtain a new structure containing values.
     //! By currying the function(s) inside the structure, it is then
     //! also possible to apply n-ary functions to n structures containing
     //! values.
     //!
     //! @note
     //! This documentation does not go into much details about the nature
     //! of applicatives. However, the [Typeclassopedia][1] is a nice
     //! Haskell-oriented resource where such information can be found.
     //!
     //!
     //! Minimal complete definition
     //! ---------------------------
     //! `lift` and `ap` satisfying the laws below. An `Applicative` must
     //! also be a `Functor`.
     //!
     //!
     //! Laws
     //! ----
     //! Given an `Applicative` `F`, the following laws must be satisfied:
     //! 1. Identity\n
     //! For all objects `xs` of tag `F(A)`,
     //! @code
     //!     ap(lift<F>(id), xs) == xs
     //! @endcode
     //!
     //! 2. Composition\n
     //! For all objects `xs` of tag `F(A)` and functions-in-an-applicative
     //! @f$ fs : F(B \to C) @f$,
     //! @f$ gs : F(A \to B) @f$,
     //! @code
     //!     ap(ap(lift<F>(compose), fs, gs), xs) == ap(fs, ap(gs, xs))
     //! @endcode
     //!
     //! 3. Homomorphism\n
     //! For all objects `x` of tag `A` and functions @f$ f : A \to B @f$,
     //! @code
     //!     ap(lift<F>(f), lift<F>(x)) == lift<F>(f(x))
     //! @endcode
     //!
     //! 4. Interchange\n
     //! For all objects `x` of tag `A` and functions-in-an-applicative
     //! @f$ fs : F(A \to B) @f$,
     //! @code
     //!     ap(fs, lift<F>(x)) == ap(lift<F>(apply(-, x)), fs)
     //! @endcode
     //! where `apply(-, x)` denotes the partial application of the `apply`
     //! function from the @ref group-functional module to the `x` argument.
     //!
     //! As a consequence of these laws, the model of `Functor` for `F` will
     //! satisfy the following for all objects `xs` of tag `F(A)` and functions
     //! @f$ f : A \to B @f$:
     //! @code
     //!     transform(xs, f) == ap(lift<F>(f), xs)
     //! @endcode
     //!
     //!
     //! Refined concept
     //! ---------------
     //! 1. `Functor` (free model)\n
     //! As a consequence of the laws, any `Applicative F` can be made a
     //! `Functor` by setting
     //! @code
     //!     transform(xs, f) = ap(lift<F>(f), xs)
     //! @endcode
     //!
     //!
     //! Concrete models
     //! ---------------
     //! `hana::lazy`, `hana::optional`, `hana::tuple`
     //!
     //!
     //! @anchor applicative-transformation
     //! Structure-preserving functions
     //! ------------------------------
     //! An _applicative transformation_ is a function @f$ t : F(X) \to G(X) @f$
     //! between two Applicatives `F` and `G`, where `X` can be any tag, and
     //! which preserves the operations of an Applicative. In other words, for
     //! all objects `x` of tag `X`, functions-in-an-applicative
     //! @f$ fs : F(X \to Y) @f$ and objects `xs` of tag `F(X)`,
     //! @code
     //!     t(lift<F>(x)) == lift<G>(x)
     //!     t(ap(fs, xs)) == ap(t(fs), t(xs))
     //! @endcode
     //!
     //! [1]: https://wiki.haskell.org/Typeclassopedia#Applicative
     template <typename A>
     struct Applicative;
 }} // end namespace boost::hana
 
 #endif // !BOOST_HANA_FWD_CONCEPT_APPLICATIVE_HPP

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