Data.Type.Equality

base-4.9.0.0: 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 types
- 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
- class (~#) j k a b => (j ~~ k) a b
- 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
- type family a == b :: Bool

# The equality types

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) |
Methods id :: cat a a Source (.) :: cat b c -> cat a b -> cat a c Source
TestEquality k ((:~:) k a) |
Methods testEquality :: f a -> f b -> Maybe (((k :~: a) :~: a) b) Source
TestCoercion k ((:~:) k a) |
Methods testCoercion :: f a -> f b -> Maybe (Coercion (k :~: a) a b) Source
(~) k a b => Bounded ((:~:) k a b) |
Methods minBound :: (k :~: a) b Source maxBound :: (k :~: a) b Source
(~) k a b => Enum ((:~:) k a b) |
Methods succ :: (k :~: a) b -> (k :~: a) b Source pred :: (k :~: a) b -> (k :~: a) b Source toEnum :: Int -> (k :~: a) b Source fromEnum :: (k :~: a) b -> Int Source enumFrom :: (k :~: a) b -> [(k :~: a) b] Source enumFromThen :: (k :~: a) b -> (k :~: a) b -> [(k :~: a) b] Source enumFromTo :: (k :~: a) b -> (k :~: a) b -> [(k :~: a) b] Source enumFromThenTo :: (k :~: a) b -> (k :~: a) b -> (k :~: a) b -> [(k :~: a) b] Source
Eq ((:~:) k a b) |
Methods (==) :: (k :~: a) b -> (k :~: a) b -> Bool Source (/=) :: (k :~: a) b -> (k :~: a) b -> Bool Source
((~) (TYPE Lifted) a b, Data a) => Data ((:~:) (TYPE Lifted) a b) |
Methods gfoldl :: (forall d c. Data d => c (d -> c) -> d -> c c) -> (forall g. g -> c g) -> (TYPE Lifted :~: a) b -> c ((TYPE Lifted :~: a) b) Source gunfold :: (forall c r. Data c => c (c -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c ((TYPE Lifted :~: a) b) Source toConstr :: (TYPE Lifted :~: a) b -> Constr Source dataTypeOf :: (TYPE Lifted :~: a) b -> DataType Source dataCast1 :: Typeable (TYPE Lifted -> TYPE Lifted) t => (forall d. Data d => c (t d)) -> Maybe (c ((TYPE Lifted :~: a) b)) Source dataCast2 :: Typeable (TYPE Lifted -> TYPE Lifted -> TYPE Lifted) t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c ((TYPE Lifted :~: a) b)) Source gmapT :: (forall c. Data c => c -> c) -> (TYPE Lifted :~: a) b -> (TYPE Lifted :~: a) b Source gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> (TYPE Lifted :~: a) b -> r Source gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> (TYPE Lifted :~: a) b -> r Source gmapQ :: (forall d. Data d => d -> u) -> (TYPE Lifted :~: a) b -> [u] Source gmapQi :: Int -> (forall d. Data d => d -> u) -> (TYPE Lifted :~: a) b -> u Source gmapM :: Monad m => (forall d. Data d => d -> m d) -> (TYPE Lifted :~: a) b -> m ((TYPE Lifted :~: a) b) Source gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> (TYPE Lifted :~: a) b -> m ((TYPE Lifted :~: a) b) Source gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> (TYPE Lifted :~: a) b -> m ((TYPE Lifted :~: a) b) Source
Ord ((:~:) k a b) |
Methods compare :: (k :~: a) b -> (k :~: a) b -> Ordering Source (<) :: (k :~: a) b -> (k :~: a) b -> Bool Source (<=) :: (k :~: a) b -> (k :~: a) b -> Bool Source (>) :: (k :~: a) b -> (k :~: a) b -> Bool Source (>=) :: (k :~: a) b -> (k :~: a) b -> Bool Source max :: (k :~: a) b -> (k :~: a) b -> (k :~: a) b Source min :: (k :~: a) b -> (k :~: a) b -> (k :~: a) b Source
(~) k a b => Read ((:~:) k a b) |
Methods readsPrec :: Int -> ReadS ((k :~: a) b) Source readList :: ReadS [(k :~: a) b] Source readPrec :: ReadPrec ((k :~: a) b) Source readListPrec :: ReadPrec [(k :~: a) b] Source
Show ((:~:) k a b) |
Methods showsPrec :: Int -> (k :~: a) b -> ShowS Source show :: (k :~: a) b -> String Source showList :: [(k :~: a) b] -> ShowS Source

class (~#) j k a b => (j ~~ k) a b Source

Lifted, heterogeneous equality. By lifted, we mean that it can be bogus (deferred type error). By heterogeneous, the two types `a` and `b` might have different kinds.

# 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.

Minimal complete definition

testEquality

Methods

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

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

Instances

TestEquality k ((:~:) k a) |
Methods testEquality :: f a -> f b -> Maybe (((k :~: a) :~: a) b) 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 |
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 (==) (k1 -> k2) 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 |