base-4.20.0.0: Core data structures and operations

Data.Type.Equality

Description

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

Since: base-4.7.0.0

Synopsis
• class a ~# b => (a :: k) ~ (b :: k)
• class a ~# b => (a :: k0) ~~ (b :: k1)
• data (a :: k) :~: (b :: k) where
• Refl :: forall {k} (a :: k). a :~: a
• data (a :: k1) :~~: (b :: k2) where
• sym :: forall {k} (a :: k) (b :: k). (a :~: b) -> b :~: a
• trans :: forall {k} (a :: k) (b :: k) (c :: k). (a :~: b) -> (b :~: c) -> a :~: c
• castWith :: (a :~: b) -> a -> b
• gcastWith :: forall {k} (a :: k) (b :: k) r. (a :~: b) -> (a ~ b => r) -> r
• apply :: forall {k1} {k2} (f :: k1 -> k2) (g :: k1 -> k2) (a :: k1) (b :: k1). (f :~: g) -> (a :~: b) -> f a :~: g b
• inner :: forall {k1} {k2} (f :: k1 -> k2) (a :: k1) (g :: k1 -> k2) (b :: k1). (f a :~: g b) -> a :~: b
• outer :: forall {k1} {k2} (f :: k1 -> k2) (a :: k1) (g :: k1 -> k2) (b :: k1). (f a :~: g b) -> f :~: g
• class TestEquality (f :: k -> Type) where
• type family (a :: k) == (b :: k) :: Bool where ...

The equality types

class a ~# b => (a :: k) ~ (b :: k) infix 4 Source #

Lifted, homogeneous equality. By lifted, we mean that it can be bogus (deferred type error). By homogeneous, the two types a and b must have the same kinds.

class a ~# b => (a :: k0) ~~ (b :: k1) infix 4 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. Because ~~ can appear unexpectedly in error messages to users who do not care about the difference between heterogeneous equality ~~ and homogeneous equality ~, this is printed as ~ unless -fprint-equality-relations is set.

In 0.7.0, the fixity was set to infix 4 to match the fixity of :~~:.

data (a :: k) :~: (b :: k) 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 base-4.7.0.0

Constructors

 Refl :: forall {k} (a :: k). a :~: a

Instances

Instances details
 Category ((:~:) :: k -> k -> Type) @since base-4.7.0.0 Instance detailsDefined in GHC.Internal.Control.Category Methodsid :: forall (a :: k). a :~: a Source #(.) :: forall (b :: k) (c :: k) (a :: k). (b :~: c) -> (a :~: b) -> a :~: c Source # TestCoercion ((:~:) a :: k -> Type) @since base-4.7.0.0 Instance detailsDefined in GHC.Internal.Data.Type.Coercion MethodstestCoercion :: forall (a0 :: k) (b :: k). (a :~: a0) -> (a :~: b) -> Maybe (Coercion a0 b) Source # TestEquality ((:~:) a :: k -> Type) @since base-4.7.0.0 Instance detailsDefined in GHC.Internal.Data.Type.Equality MethodstestEquality :: forall (a0 :: k) (b :: k). (a :~: a0) -> (a :~: b) -> Maybe (a0 :~: b) Source # (a ~ b, Data a) => Data (a :~: b) @since base-4.7.0.0 Instance detailsDefined in GHC.Internal.Data.Data Methodsgfoldl :: (forall d b0. Data d => c (d -> b0) -> d -> c b0) -> (forall g. g -> c g) -> (a :~: b) -> c (a :~: b) Source #gunfold :: (forall b0 r. Data b0 => c (b0 -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (a :~: b) Source #toConstr :: (a :~: b) -> Constr Source #dataTypeOf :: (a :~: b) -> DataType Source #dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (a :~: b)) Source #dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (a :~: b)) Source #gmapT :: (forall b0. Data b0 => b0 -> b0) -> (a :~: b) -> a :~: b Source #gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> (a :~: b) -> r Source #gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> (a :~: b) -> r Source #gmapQ :: (forall d. Data d => d -> u) -> (a :~: b) -> [u] Source #gmapQi :: Int -> (forall d. Data d => d -> u) -> (a :~: b) -> u Source #gmapM :: Monad m => (forall d. Data d => d -> m d) -> (a :~: b) -> m (a :~: b) Source #gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> (a :~: b) -> m (a :~: b) Source #gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> (a :~: b) -> m (a :~: b) Source # a ~ b => Bounded (a :~: b) @since base-4.7.0.0 Instance detailsDefined in GHC.Internal.Data.Type.Equality MethodsminBound :: a :~: b Source #maxBound :: a :~: b Source # a ~ b => Enum (a :~: b) @since base-4.7.0.0 Instance detailsDefined in GHC.Internal.Data.Type.Equality Methodssucc :: (a :~: b) -> a :~: b Source #pred :: (a :~: b) -> a :~: b Source #toEnum :: Int -> a :~: b Source #fromEnum :: (a :~: b) -> Int Source #enumFrom :: (a :~: b) -> [a :~: b] Source #enumFromThen :: (a :~: b) -> (a :~: b) -> [a :~: b] Source #enumFromTo :: (a :~: b) -> (a :~: b) -> [a :~: b] Source #enumFromThenTo :: (a :~: b) -> (a :~: b) -> (a :~: b) -> [a :~: b] Source # a ~ b => Read (a :~: b) @since base-4.7.0.0 Instance detailsDefined in GHC.Internal.Data.Type.Equality MethodsreadsPrec :: Int -> ReadS (a :~: b) Source #readList :: ReadS [a :~: b] Source # Show (a :~: b) @since base-4.7.0.0 Instance detailsDefined in GHC.Internal.Data.Type.Equality MethodsshowsPrec :: Int -> (a :~: b) -> ShowS Source #show :: (a :~: b) -> String Source #showList :: [a :~: b] -> ShowS Source # Eq (a :~: b) @since base-4.7.0.0 Instance detailsDefined in GHC.Internal.Data.Type.Equality Methods(==) :: (a :~: b) -> (a :~: b) -> Bool Source #(/=) :: (a :~: b) -> (a :~: b) -> Bool Source # Ord (a :~: b) @since base-4.7.0.0 Instance detailsDefined in GHC.Internal.Data.Type.Equality Methodscompare :: (a :~: b) -> (a :~: b) -> Ordering Source #(<) :: (a :~: b) -> (a :~: b) -> Bool Source #(<=) :: (a :~: b) -> (a :~: b) -> Bool Source #(>) :: (a :~: b) -> (a :~: b) -> Bool Source #(>=) :: (a :~: b) -> (a :~: b) -> Bool Source #max :: (a :~: b) -> (a :~: b) -> a :~: b Source #min :: (a :~: b) -> (a :~: b) -> a :~: b Source #

data (a :: k1) :~~: (b :: k2) where infix 4 Source #

Kind heterogeneous propositional equality. Like :~:, a :~~: b is inhabited by a terminating value if and only if a is the same type as b.

@since base-4.10.0.0

Constructors

 HRefl :: forall {k1} (a :: k1). a :~~: a

Instances

Instances details
 Category ((:~~:) :: k -> k -> Type) @since base-4.10.0.0 Instance detailsDefined in GHC.Internal.Control.Category Methodsid :: forall (a :: k). a :~~: a Source #(.) :: forall (b :: k) (c :: k) (a :: k). (b :~~: c) -> (a :~~: b) -> a :~~: c Source # TestCoercion ((:~~:) a :: k -> Type) @since base-4.10.0.0 Instance detailsDefined in GHC.Internal.Data.Type.Coercion MethodstestCoercion :: forall (a0 :: k) (b :: k). (a :~~: a0) -> (a :~~: b) -> Maybe (Coercion a0 b) Source # TestEquality ((:~~:) a :: k -> Type) @since base-4.10.0.0 Instance detailsDefined in GHC.Internal.Data.Type.Equality MethodstestEquality :: forall (a0 :: k) (b :: k). (a :~~: a0) -> (a :~~: b) -> Maybe (a0 :~: b) Source # (Typeable i, Typeable j, Typeable a, Typeable b, a ~~ b) => Data (a :~~: b) @since base-4.10.0.0 Instance detailsDefined in GHC.Internal.Data.Data Methodsgfoldl :: (forall d b0. Data d => c (d -> b0) -> d -> c b0) -> (forall g. g -> c g) -> (a :~~: b) -> c (a :~~: b) Source #gunfold :: (forall b0 r. Data b0 => c (b0 -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (a :~~: b) Source #toConstr :: (a :~~: b) -> Constr Source #dataTypeOf :: (a :~~: b) -> DataType Source #dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (a :~~: b)) Source #dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (a :~~: b)) Source #gmapT :: (forall b0. Data b0 => b0 -> b0) -> (a :~~: b) -> a :~~: b Source #gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> (a :~~: b) -> r Source #gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> (a :~~: b) -> r Source #gmapQ :: (forall d. Data d => d -> u) -> (a :~~: b) -> [u] Source #gmapQi :: Int -> (forall d. Data d => d -> u) -> (a :~~: b) -> u Source #gmapM :: Monad m => (forall d. Data d => d -> m d) -> (a :~~: b) -> m (a :~~: b) Source #gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> (a :~~: b) -> m (a :~~: b) Source #gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> (a :~~: b) -> m (a :~~: b) Source # a ~~ b => Bounded (a :~~: b) @since base-4.10.0.0 Instance detailsDefined in GHC.Internal.Data.Type.Equality MethodsminBound :: a :~~: b Source #maxBound :: a :~~: b Source # a ~~ b => Enum (a :~~: b) @since base-4.10.0.0 Instance detailsDefined in GHC.Internal.Data.Type.Equality Methodssucc :: (a :~~: b) -> a :~~: b Source #pred :: (a :~~: b) -> a :~~: b Source #toEnum :: Int -> a :~~: b Source #fromEnum :: (a :~~: b) -> Int Source #enumFrom :: (a :~~: b) -> [a :~~: b] Source #enumFromThen :: (a :~~: b) -> (a :~~: b) -> [a :~~: b] Source #enumFromTo :: (a :~~: b) -> (a :~~: b) -> [a :~~: b] Source #enumFromThenTo :: (a :~~: b) -> (a :~~: b) -> (a :~~: b) -> [a :~~: b] Source # a ~~ b => Read (a :~~: b) @since base-4.10.0.0 Instance detailsDefined in GHC.Internal.Data.Type.Equality MethodsreadsPrec :: Int -> ReadS (a :~~: b) Source #readList :: ReadS [a :~~: b] Source # Show (a :~~: b) @since base-4.10.0.0 Instance detailsDefined in GHC.Internal.Data.Type.Equality MethodsshowsPrec :: Int -> (a :~~: b) -> ShowS Source #show :: (a :~~: b) -> String Source #showList :: [a :~~: b] -> ShowS Source # Eq (a :~~: b) @since base-4.10.0.0 Instance detailsDefined in GHC.Internal.Data.Type.Equality Methods(==) :: (a :~~: b) -> (a :~~: b) -> Bool Source #(/=) :: (a :~~: b) -> (a :~~: b) -> Bool Source # Ord (a :~~: b) @since base-4.10.0.0 Instance detailsDefined in GHC.Internal.Data.Type.Equality Methodscompare :: (a :~~: b) -> (a :~~: b) -> Ordering Source #(<) :: (a :~~: b) -> (a :~~: b) -> Bool Source #(<=) :: (a :~~: b) -> (a :~~: b) -> Bool Source #(>) :: (a :~~: b) -> (a :~~: b) -> Bool Source #(>=) :: (a :~~: b) -> (a :~~: b) -> Bool Source #max :: (a :~~: b) -> (a :~~: b) -> a :~~: b Source #min :: (a :~~: b) -> (a :~~: b) -> a :~~: b Source #

Working with equality

sym :: forall {k} (a :: k) (b :: k). (a :~: b) -> b :~: a Source #

Symmetry of equality

trans :: forall {k} (a :: k) (b :: k) (c :: k). (a :~: b) -> (b :~: c) -> a :~: c Source #

Transitivity of equality

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

Type-safe cast, using propositional equality

gcastWith :: forall {k} (a :: k) (b :: k) r. (a :~: b) -> (a ~ b => r) -> r Source #

Generalized form of type-safe cast using propositional equality

apply :: forall {k1} {k2} (f :: k1 -> k2) (g :: k1 -> k2) (a :: k1) (b :: k1). (f :~: g) -> (a :~: b) -> f a :~: g b Source #

Apply one equality to another, respectively

inner :: forall {k1} {k2} (f :: k1 -> k2) (a :: k1) (g :: k1 -> k2) (b :: k1). (f a :~: g b) -> a :~: b Source #

Extract equality of the arguments from an equality of applied types

outer :: forall {k1} {k2} (f :: k1 -> k2) (a :: k1) (g :: k1 -> k2) (b :: k1). (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 :: k -> Type) where Source #

This class contains types where you can learn the equality of two types from information contained in terms.

The result should be Just Refl if and only if the types applied to f are equal:

testEquality (x :: f a) (y :: f b) = Just Refl ⟺ a = b

Typically, only singleton types should inhabit this class. In that case type argument equality coincides with term equality:

testEquality (x :: f a) (y :: f b) = Just Refl ⟺ a = b ⟺ x = y
isJust (testEquality x y) = x == y

Singleton types are not required, however, and so the latter two would-be laws are not in fact valid in general.

Methods

testEquality :: forall (a :: k) (b :: k). f a -> f b -> Maybe (a :~: b) Source #

Conditionally prove the equality of a and b.

Instances

Instances details
 @since base-4.18.0.0 Instance detailsDefined in GHC.Internal.TypeNats MethodstestEquality :: forall (a :: Nat) (b :: Nat). SNat a -> SNat b -> Maybe (a :~: b) Source # @since base-4.18.0.0 Instance detailsDefined in GHC.Internal.TypeLits MethodstestEquality :: forall (a :: Char) (b :: Char). SChar a -> SChar b -> Maybe (a :~: b) Source # @since base-4.18.0.0 Instance detailsDefined in GHC.Internal.TypeLits MethodstestEquality :: forall (a :: Symbol) (b :: Symbol). SSymbol a -> SSymbol b -> Maybe (a :~: b) Source # TestEquality (TypeRep :: k -> Type) Instance detailsDefined in GHC.Internal.Data.Typeable.Internal MethodstestEquality :: forall (a :: k) (b :: k). TypeRep a -> TypeRep b -> Maybe (a :~: b) Source # TestEquality ((:~:) a :: k -> Type) @since base-4.7.0.0 Instance detailsDefined in GHC.Internal.Data.Type.Equality MethodstestEquality :: forall (a0 :: k) (b :: k). (a :~: a0) -> (a :~: b) -> Maybe (a0 :~: b) Source # TestEquality ((:~~:) a :: k -> Type) @since base-4.10.0.0 Instance detailsDefined in GHC.Internal.Data.Type.Equality MethodstestEquality :: forall (a0 :: k) (b :: k). (a :~~: a0) -> (a :~~: b) -> Maybe (a0 :~: b) Source # TestEquality f => TestEquality (Compose f g :: k2 -> Type) Source # The deduction (via generativity) that if g x :~: g y then x :~: y.Since: base-4.14.0.0 Instance detailsDefined in Data.Functor.Compose MethodstestEquality :: forall (a :: k2) (b :: k2). Compose f g a -> Compose f g b -> Maybe (a :~: b) Source #

Boolean type-level equality

type family (a :: k) == (b :: k) :: Bool where ... infix 4 Source #

A type family to compute Boolean equality.

Equations

 (f a :: k2) == (g b :: k2) = (f == g) && (a == b) (a :: k) == (a :: k) = 'True (_1 :: k) == (_2 :: k) = 'False