| Copyright | (C) 2011-2014 Edward Kmett |
|---|---|
| License | BSD-style (see the file LICENSE) |
| Maintainer | Edward Kmett <[email protected]> |
| Stability | provisional |
| Portability | rank2 types, type operators, (optional) type families |
| Safe Haskell | None |
| Language | Haskell98 |
Data.Eq.Type
Contents
Description
Leibnizian equality. Injectivity in the presence of type families is provided by a generalization of a trick by Oleg Kiselyov posted here:
http://www.haskell.org/pipermail/haskell-cafe/2010-May/077177.html
- newtype a := b = Refl {
- subst :: forall c. c a -> c b
- refl :: a := a
- trans :: (a := b) -> (b := c) -> a := c
- symm :: (a := b) -> b := a
- coerce :: (a := b) -> a -> b
- lift :: (a := b) -> f a := f b
- lift2 :: (a := b) -> f a c := f b c
- lift2' :: (a := b) -> (c := d) -> f a c := f b d
- lift3 :: (a := b) -> f a c d := f b c d
- lift3' :: (a := b) -> (c := d) -> (e := f) -> g a c e := g b d f
- lower :: forall a b f. (f a := f b) -> a := b
- lower2 :: forall a b c f. (f a c := f b c) -> a := b
- lower3 :: forall a b c d f. (f a c d := f b c d) -> a := b
- fromLeibniz :: (a := b) -> a :~: b
- toLeibniz :: (a :~: b) -> a := b
- reprLeibniz :: (a := b) -> Coercion a b
Leibnizian equality
newtype a := b infixl 4 Source #
Leibnizian equality states that two things are equal if you can substitute one for the other in all contexts
Equality as an equivalence relation
Lifting equality
Lowering equality
lower :: forall a b f. (f a := f b) -> a := b Source #
Type constructors are injective, so you can lower equality through any type constructor ...
lower3 :: forall a b c d f. (f a c d := f b c d) -> a := b Source #
... these definitions are poly-kinded on GHC 7.6 and up.
:~: equivalence
Data.Type.Equality GADT definition is equivalent in power
fromLeibniz :: (a := b) -> a :~: b Source #
Coercion conversion
Leibnizian equality can be converted to representational equality
reprLeibniz :: (a := b) -> Coercion a b Source #