```{-# LANGUAGE Rank2Types, TypeOperators #-}
-----------------------------------------------------------------------------
-- |
-- Module      :  Data.Type.Eq
-- Copyright   :  (C) 2011 Edward Kmett
--
-- Maintainer  :  Edward Kmett <ekmett@gmail.com>
-- Stability   :  provisional
-- Portability :  rank2 types, type operators, type families (optional)
--
-- Leibnizian equality. Injectivity in the presence of type families
-- is provided by a generalization of a trick by Oleg Kiselyv posted here:
--
----------------------------------------------------------------------------

module Data.Eq.Type
(
-- * Leibnizian equality
(:=)(..)
-- * Equality as an equivalence relation
, refl
, trans
, symm
, coerce
-- * Lifting equality
, lift
, lift2, lift2'
, lift3, lift3'
#ifdef LANGUAGE_TypeFamilies
-- * Lowering equality
, lower
, lower2
, lower3
#endif
) where

import Prelude ()
import Control.Category

infixl 4 :=

-- | Leibnizian equality states that two things are equal if you can
-- substite one for the other in all contexts
data a := b = Refl { subst :: forall c. c a -> c b }

-- | Equality is reflexive
refl :: a := a
refl = Refl id

newtype Coerce a = Coerce { uncoerce :: a }

-- | If two things are equal you can convert one to the other
coerce :: a := b -> a -> b
coerce f = uncoerce . subst f . Coerce

-- | Equality forms a category
instance Category (:=) where
id = Refl id
(.) = subst

-- | Equality is transitive
trans :: a := b -> b := c -> a := c
trans = (>>>)

newtype Symm p a b = Symm { unsymm :: p b a }
-- | Equality is symmetric
symm :: (a := b) -> (b := a)
symm a = unsymm (subst a (Symm id))

newtype Lift f a b = Lift { unlift :: f a := f b }
-- | You can lift equality into any type constructor
lift :: a := b -> f a := f b
lift a = unlift (subst a (Lift id))

newtype Lift2 f c a b = Lift2 { unlift2 :: f a c := f b c }
-- | ... in any position
lift2 :: a := b -> f a c := f b c
lift2 a = unlift2 (subst a (Lift2 id))

lift2' :: a := b -> c := d -> f a c := f b d
lift2' ab cd = lift2 ab . lift cd

newtype Lift3 f c d a b = Lift3 { unlift3 :: f a c d := f b c d }
lift3 :: a := b -> f a c d := f b c d
lift3 a = unlift3 (subst a (Lift3 id))

lift3' :: a := b -> c := d -> e := f -> g a c e := g b d f
lift3' ab cd ef = lift3 ab . lift2 cd . lift ef

#ifdef LANGUAGE_TypeFamilies

type family Inj f :: *
type instance Inj (f a) = a
newtype Lower a b = Lower { unlower :: Inj a := Inj b }
-- | Type constructors are injective, so you can lower equality through any type constructor
lower :: f a := f b -> a := b
lower eq = unlower (subst eq (Lower id :: Lower (f a) (f a)))

type family Inj2 f :: *
type instance Inj2 (f a b) = a
newtype Lower2 a b = Lower2 { unlower2 :: Inj2 a := Inj2 b }
-- | ... in any position
lower2 :: f a c := f b c -> a := b
lower2 eq = unlower2 (subst eq (Lower2 id :: Lower2 (f a c) (f a c)))

type family Inj3 f :: *
type instance Inj3 (f a b c) = a
newtype Lower3 a b = Lower3 { unlower3 :: Inj3 a := Inj3 b }
lower3 :: f a c d := f b c d -> a := b
lower3 eq = unlower3 (subst eq (Lower3 id :: Lower3 (f a c d) (f a c d)))

#endif
```