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/*! @file Forward declares `boost::hana::Orderable`. 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_ORDERABLE_HPP #define BOOST_HANA_FWD_CONCEPT_ORDERABLE_HPP #include namespace boost { namespace hana { //! @ingroup group-concepts //! @defgroup group-Orderable Orderable //! The `Orderable` concept represents totally ordered data types. //! //! Intuitively, `Orderable` objects must define a binary predicate named //! `less` returning whether the first argument is to be considered less //! than the second argument. The word "total" means that _distinct_ //! objects must always be ordered; if `a` and `b` are not equal, then //! exactly one of `less(a, b)` and `less(b, a)` must be true. This is //! a contrast with weaker kinds of orders that would allow some objects //! to be incomparable (neither less than nor greater than). Also note //! that a non-strict total order may always be obtained from a strict //! total order (and vice-versa) by setting //! @code //! a <= b = !(b < a) //! a < b = !(b <= a) //! @endcode //! The non-strict version is used in the description of the laws because //! it makes them easier to parse for humans, but they could be formulated //! equivalently using the strict order. //! //! //! Minimal complete definition //! --------------------------- //! `less` //! //! When `less` is defined, the other methods are defined from it using //! the same definition as mandated in the laws below. //! //! //! Laws //! ---- //! Rigorously speaking, a [total order][1] `<=` on a set `S` is a binary //! predicate @f$ <= \;: S \times S \to bool @f$ such that for all //! `a`, `b`, `c` in `S`, //! @code //! if a <= b and b <= a then a == b // Antisymmetry //! if a <= b and b <= c then a <= c // Transitivity //! either a <= b or b <= a // Totality //! @endcode //! Additionally, the `less`, `greater` and `greater_equal` methods should //! have the following intuitive meanings: //! @code //! a < b if and only if !(b <= a) //! a > b if and only if b < a //! a >= b if and only if !(a < b) //! @endcode //! //! //! Refined concept //! --------------- //! 1. `Comparable` (free model)\n //! Since `Orderable` requires `less_equal` to be a total order, a model //! of `Comparable` may always be obtained by setting //! @code //! equal(x, y) = less_equal(x, y) && less_equal(y, x) //! @endcode //! //! //! Concrete models //! --------------- //! `hana::integral_constant`, `hana::optional`, `hana::pair`, //! `hana::string`, `hana::tuple` //! //! //! Free model for `LessThanComparable` data types //! ---------------------------------------------- //! Two data types `T` and `U` that model the cross-type version of the //! usual [LessThanComparable][2] C++ concept are automatically a model //! of `Orderable` by setting //! @code //! less(x, y) = (x < y) //! @endcode //! The cross-type version of the LessThanComparable concept is analogous //! to the cross-type version of the EqualityComparable concept presented //! in [N3351][3], which is compatible with the usual single type //! definition. //! However, note that the LessThanComparable concept only requires `<` //! to be a [strict weak ordering][4], which is a weaker requirement //! than being a total order. Hence, if `less` is used with objects //! of a LessThanComparable data type that do not define a total order, //! some algorithms may have an unexpected behavior. It is the author's //! opinion that defining `operator<` as a non-total order is a bad idea, //! but this is debatable and so the design choice of providing a model //! for LessThanComparable data types is open to debate. Waiting for //! some user input. //! //! //! Order-preserving functions //! -------------------------- //! Let `A` and `B` be two `Orderable` data types. A function //! @f$ f : A \to B@f$ is said to be order-preserving (also called //! monotone) if it preserves the structure of the `Orderable` concept, //! which can be rigorously stated as follows. For all objects `x`, `y` //! of data type `A`, //! @code //! if less(x, y) then less(f(x), f(y)) //! @endcode //! Another important property is that of being order-reflecting, which //! can be stated as //! @code //! if less(f(x), f(y)) then less(x, y) //! @endcode //! We say that a function is an order-embedding if it is both //! order-preserving and order-reflecting, i.e. if //! @code //! less(x, y) if and only if less(f(x), f(y)) //! @endcode //! //! //! Cross-type version of the methods //! --------------------------------- //! The comparison methods (`less`, `less_equal`, `greater` and //! `greater_equal`) are "overloaded" to handle distinct data types //! with certain properties. Specifically, they are defined for //! _distinct_ data types `A` and `B` such that //! 1. `A` and `B` share a common data type `C`, as determined by the //! `common` metafunction //! 2. `A`, `B` and `C` are all `Orderable` when taken individually //! 3. @f$\mathrm{to} : A \to C@f$ and @f$\mathrm{to} : B \to C@f$ //! are both order-embeddings as determined by the `is_embedding` //! metafunction. //! //! The method definitions for data types satisfying the above //! properties are //! @code //! less(x, y) = less(to(x), to(y)) //! less_equal(x, y) = less_equal(to(x), to(y)) //! greater_equal(x, y) = greater_equal(to(x), to(y)) //! greater(x, y) = greater(to(x), to(y)) //! @endcode //! //! //! Partial application of the methods //! ---------------------------------- //! The `less`, `greater`, `less_equal` and `greater_equal` methods can //! be called in two different ways. First, they can be called like //! normal functions: //! @code //! less(x, y) //! greater(x, y) //! //! less_equal(x, y) //! greater_equal(x, y) //! @endcode //! //! However, they may also be partially applied to an argument as follows: //! @code //! less.than(x)(y) == less(y, x) //! greater.than(x)(y) == greater(y, x) //! //! less_equal.than(x)(y) == less_equal(y, x) //! greater_equal.than(x)(y) == greater_equal(y, x) //! @endcode //! //! Take good note that the order of the arguments is reversed, so //! for example `less.than(x)(y)` is equivalent to `less(y, x)`, not //! `less(x, y)`. This is because those variants are meant to be used //! with higher order algorithms, where the chosen application order //! makes sense. //! //! //! [1]: http://en.wikipedia.org/wiki/Total_order //! [2]: http://en.cppreference.com/w/cpp/named_req/LessThanComparable //! [3]: http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2012/n3351.pdf //! [4]: http://en.wikipedia.org/wiki/Strict_weak_ordering template struct Orderable; }} // end namespace boost::hana #endif // !BOOST_HANA_FWD_CONCEPT_ORDERABLE_HPP