base-4.7.0.0: Basic libraries

License | BSD-style (see the LICENSE file in the distribution)
Maintainer | libraries@haskell.org
Stability | experimental
Portability | not portable
Safe Haskell | None
Language | Haskell2010

Data.Type.Equality

Contents

- The equality type
- Working with equality
- Inferring equality from other types
- Boolean type-level 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

- data a :~: b where
  - Refl :: a :~: a
- sym :: (a :~: b) -> b :~: a
- trans :: (a :~: b) -> (b :~: c) -> a :~: c
- castWith :: (a :~: b) -> a -> b
- gcastWith :: (a :~: b) -> ((a ~ b) => r) -> r
- apply :: (f :~: g) -> (a :~: b) -> f a :~: g b
- inner :: (f a :~: g b) -> a :~: b
- outer :: (f a :~: g b) -> f :~: g
- class TestEquality f where
  - testEquality :: f a -> f b -> Maybe (a :~: b)
- type family a == b :: Bool

# The equality type

data a :~: b where infix 4

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

# Working with equality

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

Symmetry of equality

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

Transitivity of equality

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

Type-safe cast, using propositional equality

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

Generalized form of type-safe cast using propositional equality

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

Apply one equality to another, respectively

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

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

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

Extract equality of type constructors from an equality of applied types

# Inferring equality from other types

class TestEquality f where

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)

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

Instances

TestEquality k ((:~:) k a) |

# Boolean type-level equality

type family a == b :: Bool infix 4

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