# 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

## 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) TestEquality k ((:~:) k a) TestCoercion k ((:~:) k a) (~) 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 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)

## 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 (==) (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

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