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.
lift
and ap
satisfying the laws below. An Applicative
must also be a Functor
.
Given an Applicative
F
, the following laws must be satisfied:
xs
of tag F(A)
, xs
of tag F(A)
and functions-in-an-applicative \( fs : F(B \to C) \), \( gs : F(A \to B) \), x
of tag A
and functions \( f : A \to B \), x
of tag A
and functions-in-an-applicative \( fs : F(A \to B) \), where apply(-, x)
denotes the partial application of the apply
function from the 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 \( f : A \to B \):
Functor
(free model)Applicative F
can be made a Functor
by setting hana::lazy
, hana::optional
, hana::tuple
An applicative transformation is a function \( t : F(X) \to G(X) \) 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 \( fs : F(X \to Y) \) and objects xs
of tag F(X)
,
Variables | |
constexpr auto | boost::hana::ap |
Lifted application.Specifically, ap applies a structure containing functions to a structure containing values, and returns a new structure containing values. The exact way in which the functions are applied to the values depends on the Applicative . More... | |
template<typename A > | |
constexpr auto | boost::hana::lift |
Lift a value into an Applicative structure.lift<A> takes a normal value and embeds it into a structure whose shape is represented by the A Applicative . Note that the value may be a function, in which case the created structure may be ap plied to another Applicative structure containing values. More... | |
constexpr auto boost::hana::ap |
#include <boost/hana/fwd/ap.hpp>
Lifted application.Specifically, ap
applies a structure containing functions to a structure containing values, and returns a new structure containing values. The exact way in which the functions are applied to the values depends on the Applicative
.
ap
can be called with two arguments or more; the functions in the f
structure are curried and then applied to the values in each x...
structure using the binary form of ap
. Note that this requires the number of x...
must match the arity of the functions in the f
structure. In other words, ap(f, x1, ..., xN)
is equivalent to
where f'
is f
but containing curried functions instead and x <ap> y
is just ap(x, y)
written in infix notation to emphasize the left associativity.
Given an Applicative A
, the signature is \( \mathtt{ap} : A(T_1 \times \cdots \times T_n \to U) \times A(T_1) \times \cdots \times A(T_n) \to A(U) \).
f | A structure containing function(s). |
x... | Structure(s) containing value(s) and on which f is applied. The number of structures must match the arity of the functions in the f structure. |
constexpr auto boost::hana::lift |
#include <boost/hana/fwd/lift.hpp>
Lift a value into an Applicative
structure.lift<A>
takes a normal value and embeds it into a structure whose shape is represented by the A
Applicative
. Note that the value may be a function, in which case the created structure may be ap
plied to another Applicative
structure containing values.
Given an Applicative A
, the signature is \( \mathtt{lift}_A : T \to A(T) \).
A | A tag representing the Applicative into which the value is lifted. |
x | The value to lift into the applicative. |