ceph/src/boost/libs/hana/include/boost/hana/fwd/concept/applicative.hpp

   1 /*!
   2 @file
   3 Forward declares `boost::hana::Applicative`.
   4
   5 @copyright Louis Dionne 2013-2016
   6 Distributed under the Boost Software License, Version 1.0.
   7 (See accompanying file LICENSE.md or copy at http://boost.org/LICENSE_1_0.txt)
   8  */
   9
  10 #ifndef BOOST_HANA_FWD_CONCEPT_APPLICATIVE_HPP
  11 #define BOOST_HANA_FWD_CONCEPT_APPLICATIVE_HPP
  12
  13 #include <boost/hana/config.hpp>
  14
  15
  16 BOOST_HANA_NAMESPACE_BEGIN
  17     //! @ingroup group-concepts
  18     //! @defgroup group-Applicative Applicative
  19     //! The `Applicative` concept represents `Functor`s with the ability
  20     //! to lift values and combine computations.
  21     //!
  22     //! A `Functor` can only take a normal function and map it over a
  23     //! structure containing values to obtain a new structure containing
  24     //! values. Intuitively, an `Applicative` can also take a value and
  25     //! lift it into the structure. In addition, an `Applicative` can take
  26     //! a structure containing functions and apply it to a structure
  27     //! containing values to obtain a new structure containing values.
  28     //! By currying the function(s) inside the structure, it is then
  29     //! also possible to apply n-ary functions to n structures containing
  30     //! values.
  31     //!
  32     //! @note
  33     //! This documentation does not go into much details about the nature
  34     //! of applicatives. However, the [Typeclassopedia][1] is a nice
  35     //! Haskell-oriented resource where such information can be found.
  36     //!
  37     //!
  38     //! Minimal complete definition
  39     //! ---------------------------
  40     //! `lift` and `ap` satisfying the laws below. An `Applicative` must
  41     //! also be a `Functor`.
  42     //!
  43     //!
  44     //! Laws
  45     //! ----
  46     //! Given an `Applicative` `F`, the following laws must be satisfied:
  47     //! 1. Identity\n
  48     //! For all objects `xs` of tag `F(A)`,
  49     //! @code
  50     //!     ap(lift<F>(id), xs) == xs
  51     //! @endcode
  52     //!
  53     //! 2. Composition\n
  54     //! For all objects `xs` of tag `F(A)` and functions-in-an-applicative
  55     //! @f$ fs : F(B \to C) @f$,
  56     //! @f$ gs : F(A \to B) @f$,
  57     //! @code
  58     //!     ap(ap(lift<F>(compose), fs, gs), xs) == ap(fs, ap(gs, xs))
  59     //! @endcode
  60     //!
  61     //! 3. Homomorphism\n
  62     //! For all objects `x` of tag `A` and functions @f$ f : A \to B @f$,
  63     //! @code
  64     //!     ap(lift<F>(f), lift<F>(x)) == lift<F>(f(x))
  65     //! @endcode
  66     //!
  67     //! 4. Interchange\n
  68     //! For all objects `x` of tag `A` and functions-in-an-applicative
  69     //! @f$ fs : F(A \to B) @f$,
  70     //! @code
  71     //!     ap(fs, lift<F>(x)) == ap(lift<F>(apply(-, x)), fs)
  72     //! @endcode
  73     //! where `apply(-, x)` denotes the partial application of the `apply`
  74     //! function from the @ref group-functional module to the `x` argument.
  75     //!
  76     //! As a consequence of these laws, the model of `Functor` for `F` will
  77     //! satisfy the following for all objects `xs` of tag `F(A)` and functions
  78     //! @f$ f : A \to B @f$:
  79     //! @code
  80     //!     transform(xs, f) == ap(lift<F>(f), xs)
  81     //! @endcode
  82     //!
  83     //!
  84     //! Refined concept
  85     //! ---------------
  86     //! 1. `Functor` (free model)\n
  87     //! As a consequence of the laws, any `Applicative F` can be made a
  88     //! `Functor` by setting
  89     //! @code
  90     //!     transform(xs, f) = ap(lift<F>(f), xs)
  91     //! @endcode
  92     //!
  93     //!
  94     //! Concrete models
  95     //! ---------------
  96     //! `hana::lazy`, `hana::optional`, `hana::tuple`
  97     //!
  98     //!
  99     //! @anchor applicative-transformation
 100     //! Structure-preserving functions
 101     //! ------------------------------
 102     //! An _applicative transformation_ is a function @f$ t : F(X) \to G(X) @f$
 103     //! between two Applicatives `F` and `G`, where `X` can be any tag, and
 104     //! which preserves the operations of an Applicative. In other words, for
 105     //! all objects `x` of tag `X`, functions-in-an-applicative
 106     //! @f$ fs : F(X \to Y) @f$ and objects `xs` of tag `F(X)`,
 107     //! @code
 108     //!     t(lift<F>(x)) == lift<G>(x)
 109     //!     t(ap(fs, xs)) == ap(t(fs), t(xs))
 110     //! @endcode
 111     //!
 112     //! [1]: https://wiki.haskell.org/Typeclassopedia#Applicative
 113     template <typename A>
 114     struct Applicative;
 115 BOOST_HANA_NAMESPACE_END
 116
 117 #endif // !BOOST_HANA_FWD_CONCEPT_APPLICATIVE_HPP