102 lines
4.1 KiB
C++
102 lines
4.1 KiB
C++
/*!
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@file
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Forward declares `boost::hana::Monoid`.
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@copyright Louis Dionne 2013-2016
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE.md or copy at http://boost.org/LICENSE_1_0.txt)
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*/
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#ifndef BOOST_HANA_FWD_CONCEPT_MONOID_HPP
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#define BOOST_HANA_FWD_CONCEPT_MONOID_HPP
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#include <boost/hana/config.hpp>
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BOOST_HANA_NAMESPACE_BEGIN
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//! @ingroup group-concepts
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//! @defgroup group-Monoid Monoid
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//! The `Monoid` concept represents data types with an associative
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//! binary operation that has an identity.
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//!
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//! Specifically, a [Monoid][1] is a basic algebraic structure typically
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//! used in mathematics to construct more complex algebraic structures
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//! like `Group`s, `Ring`s and so on. They are useful in several contexts,
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//! notably to define the properties of numbers in a granular way. At its
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//! core, a `Monoid` is a set `S` of objects along with a binary operation
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//! (let's say `+`) that is associative and that has an identity in `S`.
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//! There are many examples of `Monoid`s:
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//! - strings with concatenation and the empty string as the identity
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//! - integers with addition and `0` as the identity
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//! - integers with multiplication and `1` as the identity
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//! - many others...
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//!
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//! As you can see with the integers, there are some sets that can be
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//! viewed as a monoid in more than one way, depending on the choice
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//! of the binary operation and identity. The method names used here
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//! refer to the monoid of integers under addition; `plus` is the binary
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//! operation and `zero` is the identity element of that operation.
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//!
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//!
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//! Minimal complete definition
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//! ---------------------------
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//! `plus` and `zero` satisfying the laws
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//!
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//!
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//! Laws
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//! ----
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//! For all objects `x`, `y` and `z` of a `Monoid` `M`, the following
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//! laws must be satisfied:
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//! @code
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//! plus(zero<M>(), x) == x // left zero
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//! plus(x, zero<M>()) == x // right zero
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//! plus(x, plus(y, z)) == plus(plus(x, y), z) // associativity
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//! @endcode
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//!
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//!
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//! Concrete models
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//! ---------------
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//! `hana::integral_constant`
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//!
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//!
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//! Free model for non-boolean arithmetic data types
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//! ------------------------------------------------
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//! A data type `T` is arithmetic if `std::is_arithmetic<T>::%value` is
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//! true. For a non-boolean arithmetic data type `T`, a model of `Monoid`
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//! is automatically defined by setting
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//! @code
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//! plus(x, y) = (x + y)
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//! zero<T>() = static_cast<T>(0)
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//! @endcode
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//!
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//! > #### Rationale for not making `bool` a `Monoid` by default
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//! > First, it makes no sense whatsoever to define an additive `Monoid`
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//! > over the `bool` type. Also, it could make sense to define a `Monoid`
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//! > with logical conjunction or disjunction. However, C++ allows `bool`s
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//! > to be added, and the method names of this concept really suggest
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//! > addition. In line with the principle of least surprise, no model
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//! > is provided by default.
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//!
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//!
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//! Structure-preserving functions
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//! ------------------------------
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//! Let `A` and `B` be two `Monoid`s. A function `f : A -> B` is said
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//! to be a [Monoid morphism][2] if it preserves the monoidal structure
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//! between `A` and `B`. Rigorously, for all objects `x, y` of data
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//! type `A`,
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//! @code
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//! f(plus(x, y)) == plus(f(x), f(y))
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//! f(zero<A>()) == zero<B>()
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//! @endcode
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//! Functions with these properties interact nicely with `Monoid`s, which
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//! is why they are given such a special treatment.
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//!
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//!
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//! [1]: http://en.wikipedia.org/wiki/Monoid
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//! [2]: http://en.wikipedia.org/wiki/Monoid#Monoid_homomorphisms
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template <typename M>
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struct Monoid;
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BOOST_HANA_NAMESPACE_END
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#endif // !BOOST_HANA_FWD_CONCEPT_MONOID_HPP
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