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/*!
@file
Forward declares `boost::hana::Applicative`.
@copyright Louis Dionne 2013-2016
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>
BOOST_HANA_NAMESPACE_BEGIN
//! @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;
BOOST_HANA_NAMESPACE_END
#endif // !BOOST_HANA_FWD_CONCEPT_APPLICATIVE_HPP
|
