base-4.8.0.0: Basic libraries

Data.Type.Equality

Description

Definition of propositional equality `(:~:)`. Pattern-matching on a variable of type `(a :~: b)` produces a proof that `a ~ b`.

Since: 4.7.0.0

Synopsis

# The equality type

data a :~: b where infix 4 Source

Propositional equality. If `a :~: b` is inhabited by some terminating value, then the type `a` is the same as the type `b`. To use this equality in practice, pattern-match on the `a :~: b` to get out the `Refl` constructor; in the body of the pattern-match, the compiler knows that `a ~ b`.

Since: 4.7.0.0

Constructors

 Refl :: a :~: a

Instances

 Category k ((:~:) k) Source TestEquality k ((:~:) k a) Source TestCoercion k ((:~:) k a) Source (~) k a b => Bounded ((:~:) k a b) Source (~) k a b => Enum ((:~:) k a b) Source Eq ((:~:) k a b) Source ((~) * a b, Data a) => Data ((:~:) * a b) Source Ord ((:~:) k a b) Source (~) k a b => Read ((:~:) k a b) Source Show ((:~:) k a b) Source

# Working with equality

sym :: (a :~: b) -> b :~: a Source

Symmetry of equality

trans :: (a :~: b) -> (b :~: c) -> a :~: c Source

Transitivity of equality

castWith :: (a :~: b) -> a -> b Source

Type-safe cast, using propositional equality

gcastWith :: (a :~: b) -> ((a ~ b) => r) -> r Source

Generalized form of type-safe cast using propositional equality

apply :: (f :~: g) -> (a :~: b) -> f a :~: g b Source

Apply one equality to another, respectively

inner :: (f a :~: g b) -> a :~: b Source

Extract equality of the arguments from an equality of a applied types

outer :: (f a :~: g b) -> f :~: g Source

Extract equality of type constructors from an equality of applied types

# Inferring equality from other types

class TestEquality f where Source

This class contains types where you can learn the equality of two types from information contained in terms. Typically, only singleton types should inhabit this class.

Methods

testEquality :: f a -> f b -> Maybe (a :~: b) Source

Conditionally prove the equality of `a` and `b`.

Instances

 TestEquality k ((:~:) k a) Source

# Boolean type-level equality

type family a == b :: Bool infix 4 Source

A type family to compute Boolean equality. Instances are provided only for open kinds, such as `*` and function kinds. Instances are also provided for datatypes exported from base. A poly-kinded instance is not provided, as a recursive definition for algebraic kinds is generally more useful.

Instances

 type (==) Bool a b Source type (==) Ordering a b Source type (==) * a b Source type (==) Nat a b Source type (==) Symbol a b Source type (==) () a b Source type (==) [k] a b Source type (==) (Maybe k) a b Source type (==) (k -> k1) a b Source type (==) (Either k k1) a b Source type (==) ((,) k k1) a b Source type (==) ((,,) k k1 k2) a b Source type (==) ((,,,) k k1 k2 k3) a b Source type (==) ((,,,,) k k1 k2 k3 k4) a b Source type (==) ((,,,,,) k k1 k2 k3 k4 k5) a b Source type (==) ((,,,,,,) k k1 k2 k3 k4 k5 k6) a b Source type (==) ((,,,,,,,) k k1 k2 k3 k4 k5 k6 k7) a b Source type (==) ((,,,,,,,,) k k1 k2 k3 k4 k5 k6 k7 k8) a b Source type (==) ((,,,,,,,,,) k k1 k2 k3 k4 k5 k6 k7 k8 k9) a b Source type (==) ((,,,,,,,,,,) k k1 k2 k3 k4 k5 k6 k7 k8 k9 k10) a b Source type (==) ((,,,,,,,,,,,) k k1 k2 k3 k4 k5 k6 k7 k8 k9 k10 k11) a b Source type (==) ((,,,,,,,,,,,,) k k1 k2 k3 k4 k5 k6 k7 k8 k9 k10 k11 k12) a b Source type (==) ((,,,,,,,,,,,,,) k k1 k2 k3 k4 k5 k6 k7 k8 k9 k10 k11 k12 k13) a b Source type (==) ((,,,,,,,,,,,,,,) k k1 k2 k3 k4 k5 k6 k7 k8 k9 k10 k11 k12 k13 k14) a b Source