{-# LANGUAGE DeriveGeneric        #-}
{-# LANGUAGE TypeOperators        #-}
{-# LANGUAGE GADTs                #-}
{-# LANGUAGE FlexibleInstances    #-}
{-# LANGUAGE StandaloneDeriving   #-}
{-# LANGUAGE NoImplicitPrelude    #-}
{-# LANGUAGE PolyKinds            #-}
{-# LANGUAGE RankNTypes           #-}
{-# LANGUAGE DataKinds            #-}
{-# LANGUAGE TypeFamilies         #-}
{-# LANGUAGE UndecidableInstances #-}
{-# LANGUAGE ExplicitNamespaces   #-}

-----------------------------------------------------------------------------
-- |
-- Module      :  Data.Type.Equality
-- License     :  BSD-style (see the LICENSE file in the distribution)
--
-- Maintainer  :  libraries@haskell.org
-- Stability   :  experimental
-- Portability :  not portable
--
-- Definition of propositional equality @(:~:)@. Pattern-matching on a variable
-- of type @(a :~: b)@ produces a proof that @a ~ b@.
--
-- /Since: 4.7.0.0/
-----------------------------------------------------------------------------



module Data.Type.Equality (
  -- * The equality type
  (:~:)(..),
  
  -- * Working with equality
  sym, trans, castWith, gcastWith, apply, inner, outer,

  -- * Inferring equality from other types
  TestEquality(..),

  -- * Boolean type-level equality
  type (==)
  ) where

import Data.Maybe
import GHC.Enum
import GHC.Show
import GHC.Read
import GHC.Base
import Data.Type.Bool

infix 4 :~:

-- | 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/
data a :~: b where
  Refl :: a :~: a

-- with credit to Conal Elliott for 'ty', Erik Hesselink & Martijn van
-- Steenbergen for 'type-equality', Edward Kmett for 'eq', and Gabor Greif
-- for 'type-eq'

-- | Symmetry of equality
sym :: (a :~: b) -> (b :~: a)
sym Refl = Refl

-- | Transitivity of equality
trans :: (a :~: b) -> (b :~: c) -> (a :~: c)
trans Refl Refl = Refl

-- | Type-safe cast, using propositional equality
castWith :: (a :~: b) -> a -> b
castWith Refl x = x

-- | Generalized form of type-safe cast using propositional equality
gcastWith :: (a :~: b) -> ((a ~ b) => r) -> r
gcastWith Refl x = x

-- | Apply one equality to another, respectively
apply :: (f :~: g) -> (a :~: b) -> (f a :~: g b)
apply Refl Refl = Refl

-- | Extract equality of the arguments from an equality of a applied types
inner :: (f a :~: g b) -> (a :~: b)
inner Refl = Refl

-- | Extract equality of type constructors from an equality of applied types
outer :: (f a :~: g b) -> (f :~: g)
outer Refl = Refl

deriving instance Eq   (a :~: b)
deriving instance Show (a :~: b)
deriving instance Ord  (a :~: b)

instance a ~ b => Read (a :~: b) where
  readsPrec d = readParen (d > 10) (\r -> [(Refl, s) | ("Refl",s) <- lex r ])

instance a ~ b => Enum (a :~: b) where
  toEnum 0 = Refl
  toEnum _ = error "Data.Type.Equality.toEnum: bad argument"

  fromEnum Refl = 0

instance a ~ b => Bounded (a :~: b) where
  minBound = Refl
  maxBound = Refl

-- | 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.
class TestEquality f where
  -- | Conditionally prove the equality of @a@ and @b@.
  testEquality :: f a -> f b -> Maybe (a :~: b)

instance TestEquality ((:~:) a) where
  testEquality Refl Refl = Just Refl

-- | 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.
type family (a :: k) == (b :: k) :: Bool
infix 4 ==

-- all of the following closed type families are local to this module
type family EqStar (a :: *) (b :: *) where
  EqStar a a = True
  EqStar a b = False

-- This looks dangerous, but it isn't. This allows == to be defined
-- over arbitrary type constructors.
type family EqArrow (a :: k1 -> k2) (b :: k1 -> k2) where
  EqArrow a a = True
  EqArrow a b = False

type family EqBool a b where
  EqBool True  True  = True
  EqBool False False = True
  EqBool a     b     = False

type family EqOrdering a b where
  EqOrdering LT LT = True
  EqOrdering EQ EQ = True
  EqOrdering GT GT = True
  EqOrdering a  b  = False

type EqUnit (a :: ()) (b :: ()) = True

type family EqList a b where
  EqList '[]        '[]        = True
  EqList (h1 ': t1) (h2 ': t2) = (h1 == h2) && (t1 == t2)
  EqList a          b          = False

type family EqMaybe a b where
  EqMaybe Nothing  Nothing  = True
  EqMaybe (Just x) (Just y) = x == y
  EqMaybe a        b        = False

type family Eq2 a b where
  Eq2 '(a1, b1) '(a2, b2) = a1 == a2 && b1 == b2

type family Eq3 a b where
  Eq3 '(a1, b1, c1) '(a2, b2, c2) = a1 == a2 && b1 == b2 && c1 == c2

type family Eq4 a b where
  Eq4 '(a1, b1, c1, d1) '(a2, b2, c2, d2) = a1 == a2 && b1 == b2 && c1 == c2 && d1 == d2

type family Eq5 a b where
  Eq5 '(a1, b1, c1, d1, e1) '(a2, b2, c2, d2, e2) = a1 == a2 && b1 == b2 && c1 == c2 && d1 == d2 && e1 == e2

type family Eq6 a b where
  Eq6 '(a1, b1, c1, d1, e1, f1) '(a2, b2, c2, d2, e2, f2) = a1 == a2 && b1 == b2 && c1 == c2 && d1 == d2 && e1 == e2 && f1 == f2

type family Eq7 a b where
  Eq7 '(a1, b1, c1, d1, e1, f1, g1) '(a2, b2, c2, d2, e2, f2, g2) = a1 == a2 && b1 == b2 && c1 == c2 && d1 == d2 && e1 == e2 && f1 == f2 && g1 == g2

type family Eq8 a b where
  Eq8 '(a1, b1, c1, d1, e1, f1, g1, h1) '(a2, b2, c2, d2, e2, f2, g2, h2) = a1 == a2 && b1 == b2 && c1 == c2 && d1 == d2 && e1 == e2 && f1 == f2 && g1 == g2 && h1 == h2

type family Eq9 a b where
  Eq9 '(a1, b1, c1, d1, e1, f1, g1, h1, i1) '(a2, b2, c2, d2, e2, f2, g2, h2, i2) = a1 == a2 && b1 == b2 && c1 == c2 && d1 == d2 && e1 == e2 && f1 == f2 && g1 == g2 && h1 == h2 && i1 == i2

type family Eq10 a b where
  Eq10 '(a1, b1, c1, d1, e1, f1, g1, h1, i1, j1) '(a2, b2, c2, d2, e2, f2, g2, h2, i2, j2) = a1 == a2 && b1 == b2 && c1 == c2 && d1 == d2 && e1 == e2 && f1 == f2 && g1 == g2 && h1 == h2 && i1 == i2 && j1 == j2

type family Eq11 a b where
  Eq11 '(a1, b1, c1, d1, e1, f1, g1, h1, i1, j1, k1) '(a2, b2, c2, d2, e2, f2, g2, h2, i2, j2, k2) = a1 == a2 && b1 == b2 && c1 == c2 && d1 == d2 && e1 == e2 && f1 == f2 && g1 == g2 && h1 == h2 && i1 == i2 && j1 == j2 && k1 == k2

type family Eq12 a b where
  Eq12 '(a1, b1, c1, d1, e1, f1, g1, h1, i1, j1, k1, l1) '(a2, b2, c2, d2, e2, f2, g2, h2, i2, j2, k2, l2) = a1 == a2 && b1 == b2 && c1 == c2 && d1 == d2 && e1 == e2 && f1 == f2 && g1 == g2 && h1 == h2 && i1 == i2 && j1 == j2 && k1 == k2 && l1 == l2

type family Eq13 a b where
  Eq13 '(a1, b1, c1, d1, e1, f1, g1, h1, i1, j1, k1, l1, m1) '(a2, b2, c2, d2, e2, f2, g2, h2, i2, j2, k2, l2, m2) = a1 == a2 && b1 == b2 && c1 == c2 && d1 == d2 && e1 == e2 && f1 == f2 && g1 == g2 && h1 == h2 && i1 == i2 && j1 == j2 && k1 == k2 && l1 == l2 && m1 == m2

type family Eq14 a b where
  Eq14 '(a1, b1, c1, d1, e1, f1, g1, h1, i1, j1, k1, l1, m1, n1) '(a2, b2, c2, d2, e2, f2, g2, h2, i2, j2, k2, l2, m2, n2) = a1 == a2 && b1 == b2 && c1 == c2 && d1 == d2 && e1 == e2 && f1 == f2 && g1 == g2 && h1 == h2 && i1 == i2 && j1 == j2 && k1 == k2 && l1 == l2 && m1 == m2 && n1 == n2

type family Eq15 a b where
  Eq15 '(a1, b1, c1, d1, e1, f1, g1, h1, i1, j1, k1, l1, m1, n1, o1) '(a2, b2, c2, d2, e2, f2, g2, h2, i2, j2, k2, l2, m2, n2, o2) = a1 == a2 && b1 == b2 && c1 == c2 && d1 == d2 && e1 == e2 && f1 == f2 && g1 == g2 && h1 == h2 && i1 == i2 && j1 == j2 && k1 == k2 && l1 == l2 && m1 == m2 && n1 == n2 && o1 == o2

-- these all look to be overlapping, but they are differentiated by their kinds
type instance a == b = EqStar a b
type instance a == b = EqArrow a b
type instance a == b = EqBool a b
type instance a == b = EqOrdering a b
type instance a == b = EqUnit a b
type instance a == b = EqList a b
type instance a == b = EqMaybe a b
type instance a == b = Eq2 a b
type instance a == b = Eq3 a b
type instance a == b = Eq4 a b
type instance a == b = Eq5 a b
type instance a == b = Eq6 a b
type instance a == b = Eq7 a b
type instance a == b = Eq8 a b
type instance a == b = Eq9 a b
type instance a == b = Eq10 a b
type instance a == b = Eq11 a b
type instance a == b = Eq12 a b
type instance a == b = Eq13 a b
type instance a == b = Eq14 a b
type instance a == b = Eq15 a b
