{-# LANGUAGE CPP, DataKinds, FlexibleContexts, GADTs, KindSignatures #-}
{-# LANGUAGE PolyKinds, RankNTypes, ScopedTypeVariables              #-}
{-# LANGUAGE StandaloneDeriving, TypeFamilies, TypeOperators         #-}
{-# LANGUAGE TypeSynonymInstances, UndecidableInstances              #-}
#if defined(__GLASGOW_HASKELL__) && __GLASGOW_HASKELL__ >= 800
{-# LANGUAGE ConstrainedClassMethods, TypeFamilyDependencies #-}
#endif
module Proof.Equational ( (:~:)(..), (:=:)
                        , sym, trans
                        , Equality(..), Preorder(..), reflexivity'
                        , (:\/:), (:/\:), (=<=), (=>=), (=~=), Leibniz(..)
                        , Reason(..), because, by, (===), start, byDefinition
                        , admitted, Proxy(..), cong, cong'
                        , Proposition(..), HVec(..), FromBool (..)
                        , applyNAry, applyNAry', fromBool'
                          -- * Conversion between equalities
                        , fromRefl, fromLeibniz, reflToLeibniz, leibnizToRefl
                          -- * Coercion
                        , coerce, coerce', withRefl
                          -- * Re-exported modules
                        , module Data.Proxy
                        ) where
import Data.Proxy
import Data.Type.Equality hiding (apply)
import Unsafe.Coerce

infix 4 :=:
type a :\/: b = Either a b
infixr 2 :\/:

type a :/\: b = (a, b)
infixr 3 :/\:

type (:=:) = (:~:)

data Leibniz a b = Leibniz { Leibniz a b -> forall (f :: k -> *). f a -> f b
apply :: forall f. f a -> f b }

leibnizToRefl :: Leibniz a b -> a :=: b
leibnizToRefl :: Leibniz a b -> a :=: b
leibnizToRefl Leibniz a b
eq = Leibniz a b -> (a :~: a) -> a :=: b
forall k (a :: k) (b :: k).
Leibniz a b -> forall (f :: k -> *). f a -> f b
apply Leibniz a b
eq a :~: a
forall k (a :: k). a :~: a
Refl

fromLeibniz :: (Preorder eq) => Leibniz a b -> eq a b
fromLeibniz :: Leibniz a b -> eq a b
fromLeibniz Leibniz a b
eq = Leibniz a b -> eq a a -> eq a b
forall k (a :: k) (b :: k).
Leibniz a b -> forall (f :: k -> *). f a -> f b
apply Leibniz a b
eq (Proxy a -> eq a a
forall k (eq :: k -> k -> *) (proxy :: k -> *) (a :: k).
Preorder eq =>
proxy a -> eq a a
reflexivity Proxy a
forall k (t :: k). Proxy t
Proxy)

fromRefl :: (Preorder eq) => a :=: b -> eq a b
fromRefl :: (a :=: b) -> eq a b
fromRefl a :=: b
Refl = eq a b
forall k (r :: k -> k -> *) (x :: k). Preorder r => r x x
reflexivity'

reflToLeibniz :: a :=: b -> Leibniz a b
reflToLeibniz :: (a :=: b) -> Leibniz a b
reflToLeibniz a :=: b
Refl = (forall (f :: k -> *). f a -> f b) -> Leibniz a b
forall k (a :: k) (b :: k).
(forall (f :: k -> *). f a -> f b) -> Leibniz a b
Leibniz forall a. a -> a
forall (f :: k -> *). f a -> f b
id

class Preorder (eq :: k -> k -> *) where
  reflexivity  :: proxy a -> eq a a
  transitivity :: eq a b  -> eq b c -> eq a c

class Preorder eq => Equality (eq :: k -> k -> *) where
  symmetry     :: eq a b  -> eq b a

instance Preorder (:=:) where
  {-# SPECIALISE instance Preorder (:=:) #-}
  transitivity :: (a :=: b) -> (b :=: c) -> a :=: c
transitivity a :=: b
Refl b :=: c
Refl = a :=: c
forall k (a :: k). a :~: a
Refl
  {-# INLINE[1] transitivity #-}

  reflexivity :: proxy a -> a :=: a
reflexivity  proxy a
_         = a :=: a
forall k (a :: k). a :~: a
Refl
  {-# INLINE[1] reflexivity #-}

instance Equality (:=:) where
  {-# SPECIALISE instance Equality (:~:) #-}
  symmetry :: (a :=: b) -> b :=: a
symmetry     a :=: b
Refl      = b :=: a
forall k (a :: k). a :~: a
Refl
  {-# INLINE[1] symmetry #-}

instance Preorder (->) where
  reflexivity :: proxy a -> a -> a
reflexivity proxy a
_ = a -> a
forall a. a -> a
id
  transitivity :: (a -> b) -> (b -> c) -> a -> c
transitivity = ((b -> c) -> (a -> b) -> a -> c) -> (a -> b) -> (b -> c) -> a -> c
forall a b c. (a -> b -> c) -> b -> a -> c
flip (b -> c) -> (a -> b) -> a -> c
forall b c a. (b -> c) -> (a -> b) -> a -> c
(.)

leibniz_refl :: Leibniz a a
leibniz_refl :: Leibniz a a
leibniz_refl = (forall (f :: k -> *). f a -> f a) -> Leibniz a a
forall k (a :: k) (b :: k).
(forall (f :: k -> *). f a -> f b) -> Leibniz a b
Leibniz forall a. a -> a
forall (f :: k -> *). f a -> f a
id

instance Preorder Leibniz where
  reflexivity :: proxy a -> Leibniz a a
reflexivity proxy a
_ = Leibniz a a
forall k (a :: k). Leibniz a a
leibniz_refl
  transitivity :: Leibniz a b -> Leibniz b c -> Leibniz a c
transitivity (Leibniz forall (f :: k -> *). f a -> f b
aEqb) (Leibniz forall (f :: k -> *). f b -> f c
bEqc) = (forall (f :: k -> *). f a -> f c) -> Leibniz a c
forall k (a :: k) (b :: k).
(forall (f :: k -> *). f a -> f b) -> Leibniz a b
Leibniz ((forall (f :: k -> *). f a -> f c) -> Leibniz a c)
-> (forall (f :: k -> *). f a -> f c) -> Leibniz a c
forall a b. (a -> b) -> a -> b
$ f b -> f c
forall (f :: k -> *). f b -> f c
bEqc (f b -> f c) -> (f a -> f b) -> f a -> f c
forall b c a. (b -> c) -> (a -> b) -> a -> c
. f a -> f b
forall (f :: k -> *). f a -> f b
aEqb

instance Equality Leibniz where
  symmetry :: Leibniz a b -> Leibniz b a
symmetry Leibniz a b
eq  = Flip Leibniz a b -> Leibniz b a
forall k k (f :: k -> k -> *) (a :: k) (b :: k).
Flip f a b -> f b a
unFlip (Flip Leibniz a b -> Leibniz b a)
-> Flip Leibniz a b -> Leibniz b a
forall a b. (a -> b) -> a -> b
$ Leibniz a b -> forall (f :: k -> *). f a -> f b
forall k (a :: k) (b :: k).
Leibniz a b -> forall (f :: k -> *). f a -> f b
apply Leibniz a b
eq (Flip Leibniz a a -> Flip Leibniz a b)
-> Flip Leibniz a a -> Flip Leibniz a b
forall a b. (a -> b) -> a -> b
$ Leibniz a a -> Flip Leibniz a a
forall k k (f :: k -> k -> *) (a :: k) (b :: k).
f b a -> Flip f a b
Flip Leibniz a a
forall k (a :: k). Leibniz a a
leibniz_refl

newtype Flip f a b = Flip { Flip f a b -> f b a
unFlip :: f b a }

data Reason eq x y where
  Because :: proxy y -> eq x y -> Reason eq x y

reflexivity' :: (Preorder r) => r x x
reflexivity' :: r x x
reflexivity' = Proxy x -> r x x
forall k (eq :: k -> k -> *) (proxy :: k -> *) (a :: k).
Preorder eq =>
proxy a -> eq a a
reflexivity Proxy x
forall k (t :: k). Proxy t
Proxy

by, because :: proxy y -> eq x y -> Reason eq x y
because :: proxy y -> eq x y -> Reason eq x y
because = proxy y -> eq x y -> Reason eq x y
forall k k (proxy :: k -> *) (y :: k) (eq :: k -> k -> *) (x :: k).
proxy y -> eq x y -> Reason eq x y
Because
by :: proxy y -> eq x y -> Reason eq x y
by      = proxy y -> eq x y -> Reason eq x y
forall k k (proxy :: k -> *) (y :: k) (eq :: k -> k -> *) (x :: k).
proxy y -> eq x y -> Reason eq x y
Because

infixl 4 ===, =<=, =~=, =>=
infix 5 `Because`
infix 5 `because`

(=<=) :: Preorder r => r x y -> Reason r y z -> r x z
r x y
eq =<= :: r x y -> Reason r y z -> r x z
=<= (proxy z
_ `Because` r y z
eq') = r x y -> r y z -> r x z
forall k (eq :: k -> k -> *) (a :: k) (b :: k) (c :: k).
Preorder eq =>
eq a b -> eq b c -> eq a c
transitivity r x y
eq r y z
eq'

{-# SPECIALISE INLINE[1] (=<=) :: x :~: y -> Reason (:~:) y z -> x :~: z #-}

(=>=) :: Preorder r => r y z -> Reason r x y -> r x z
r y z
eq =>= :: r y z -> Reason r x y -> r x z
=>= (proxy y
_ `Because` r x y
eq') = r x y -> r y z -> r x z
forall k (eq :: k -> k -> *) (a :: k) (b :: k) (c :: k).
Preorder eq =>
eq a b -> eq b c -> eq a c
transitivity r x y
eq' r y z
eq

{-# SPECIALISE INLINE[1] (=>=) :: y :~: z -> Reason (:~:) x y -> x :~: z #-}

(===) :: Equality eq => eq x y -> Reason eq y z -> eq x z
=== :: eq x y -> Reason eq y z -> eq x z
(===) = eq x y -> Reason eq y z -> eq x z
forall k (r :: k -> k -> *) (x :: k) (y :: k) (z :: k).
Preorder r =>
r x y -> Reason r y z -> r x z
(=<=)
{-# SPECIALISE INLINE[1] (===) :: x :~: y -> Reason (:~:) y z -> x :~: z #-}

(=~=) :: r x y -> proxy y -> r x y
r x y
eq =~= :: r x y -> proxy y -> r x y
=~= proxy y
_ = r x y
eq

start :: Preorder eq => proxy a -> eq a a
start :: proxy a -> eq a a
start = proxy a -> eq a a
forall k (eq :: k -> k -> *) (proxy :: k -> *) (a :: k).
Preorder eq =>
proxy a -> eq a a
reflexivity

byDefinition :: (Preorder eq) => eq a a
byDefinition :: eq a a
byDefinition = Proxy a -> eq a a
forall k (eq :: k -> k -> *) (proxy :: k -> *) (a :: k).
Preorder eq =>
proxy a -> eq a a
reflexivity Proxy a
forall k (t :: k). Proxy t
Proxy

admitted :: Reason eq x y
admitted :: Reason eq x y
admitted = Reason eq x y
forall a. HasCallStack => a
undefined
{-# WARNING admitted "There are some goals left yet unproven." #-}

cong :: forall f a b. Proxy f -> a :=: b -> f a :=: f b
cong :: Proxy f -> (a :=: b) -> f a :=: f b
cong Proxy f
Proxy a :=: b
Refl = f a :=: f b
forall k (a :: k). a :~: a
Refl

cong' :: (pxy m -> pxy (f m)) -> a :=: b -> f a :=: f b
cong' :: (pxy m -> pxy (f m)) -> (a :=: b) -> f a :=: f b
cong' pxy m -> pxy (f m)
_ a :=: b
Refl =  f a :=: f b
forall k (a :: k). a :~: a
Refl

-- | Type coercion. 'coerce' is using @unsafeCoerce a@.
-- So, please, please do not provide the @undefined@ as the proof.
-- Using this function instead of pattern-matching on equality proof,
-- you can reduce the overhead introduced by run-time proof.
coerce :: (a :=: b) -> f a -> f b
coerce :: (a :=: b) -> f a -> f b
coerce a :=: b
_ f a
a = f a -> f b
forall a b. a -> b
unsafeCoerce f a
a
{-# INLINE[1] coerce #-}

-- | Coercion for identity types.
coerce' :: a :=: b -> a -> b
coerce' :: (a :=: b) -> a -> b
coerce' a :=: b
_ a
a = a -> b
forall a b. a -> b
unsafeCoerce a
a
{-# INLINE[1] coerce' #-}

{-# RULES
"coerce/unsafeCoerce" [~1] forall xs.
  coerce xs = unsafeCoerce
"coerce'/unsafeCoerce" [~1] forall xs.
  coerce' xs = unsafeCoerce
  #-}

-- | Solves equality constraint without explicit coercion.
--   It has the same effect as @'Data.Type.Equality.gcastWith'@,
--   but some hacks is done to reduce runtime overhead.
withRefl :: forall a b r. a :~: b -> (a ~ b => r) -> r
withRefl :: (a :~: b) -> ((a ~ b) => r) -> r
withRefl a :~: b
_ (a ~ b) => r
r = case (() :~: ()) -> a :~: b
forall a b. a -> b
unsafeCoerce (() :~: ()
forall k (a :: k). a :~: a
Refl :: () :~: ()) :: a :~: b of
  a :~: b
Refl -> r
(a ~ b) => r
r
{-# NOINLINE withRefl #-}
{-# RULES
"withRefl/unsafeCoerce" [~1] forall x.
  withRefl x = unsafeCoerce
  #-}

class Proposition (f :: k -> *) where
  type OriginalProp (f :: k -> *) (n :: k) :: *
  unWrap :: f n -> OriginalProp f n
  wrap   :: OriginalProp f n -> f n

data HVec (xs :: [*]) where
  HNil :: HVec '[]
  (:-) :: x -> HVec xs -> HVec (x ': xs)

infixr 9 :-
type family (xs :: [*]) :~> (a :: *) :: * where
  '[]       :~> a = a
  (x ': xs) :~> a = x -> (xs :~> a)

infixr 1 :~>

data HVecView (xs :: [*]) :: * where
  HNilView  :: HVecView '[]
  HConsView :: Proxy t -> HVecView ts -> HVecView (t ': ts)

deriving instance Show (HVecView xs)

class KnownTypeList (xs :: [*]) where
  viewHVec' :: HVecView xs

instance KnownTypeList '[] where
  viewHVec' :: HVecView '[]
viewHVec' = HVecView '[]
HNilView

instance KnownTypeList ts => KnownTypeList (t ': ts) where
  viewHVec' :: HVecView (t : ts)
viewHVec' = Proxy t -> HVecView ts -> HVecView (t : ts)
forall t (ts :: [*]). Proxy t -> HVecView ts -> HVecView (t : ts)
HConsView Proxy t
forall k (t :: k). Proxy t
Proxy HVecView ts
forall (xs :: [*]). KnownTypeList xs => HVecView xs
viewHVec'

newtype Magic (xs :: [*]) a = Magic { Magic xs a -> KnownTypeList xs => a
_viewHVec' :: KnownTypeList xs => a }

withKnownTypeList :: forall a xs. HVecView xs -> (KnownTypeList xs => a) -> a
withKnownTypeList :: HVecView xs -> (KnownTypeList xs => a) -> a
withKnownTypeList HVecView xs
xs KnownTypeList xs => a
f = (Magic xs a -> HVecView xs -> a
forall a b. a -> b
unsafeCoerce ((KnownTypeList xs => a) -> Magic xs a
forall (xs :: [*]) a. (KnownTypeList xs => a) -> Magic xs a
Magic KnownTypeList xs => a
f :: Magic xs a) :: HVecView xs -> a) HVecView xs
xs

apply' :: HVecView ts -> (HVec ts -> c) -> ts :~> c
apply' :: HVecView ts -> (HVec ts -> c) -> ts :~> c
apply' HVecView ts
HNilView HVec ts -> c
f = HVec ts -> c
f HVec ts
HVec '[]
HNil
apply' (HConsView Proxy t
Proxy HVecView ts
ts) HVec ts -> c
f = \t
a -> HVecView ts -> (KnownTypeList ts => ts :~> c) -> ts :~> c
forall a (xs :: [*]). HVecView xs -> (KnownTypeList xs => a) -> a
withKnownTypeList HVecView ts
ts ((KnownTypeList ts => ts :~> c) -> ts :~> c)
-> (KnownTypeList ts => ts :~> c) -> ts :~> c
forall a b. (a -> b) -> a -> b
$
  HVecView ts -> (HVec ts -> c) -> ts :~> c
forall (ts :: [*]) c. HVecView ts -> (HVec ts -> c) -> ts :~> c
apply' HVecView ts
ts (\HVec ts
ts' -> HVec ts -> c
f (HVec ts -> c) -> HVec ts -> c
forall a b. (a -> b) -> a -> b
$ t
a t -> HVec ts -> HVec (t : ts)
forall x (xs :: [*]). x -> HVec xs -> HVec (x : xs)
:- HVec ts
ts')

applyNAry :: forall ts c. KnownTypeList ts => (HVec ts -> c) -> ts :~> c
applyNAry :: (HVec ts -> c) -> ts :~> c
applyNAry = HVecView ts -> (HVec ts -> c) -> ts :~> c
forall (ts :: [*]) c. HVecView ts -> (HVec ts -> c) -> ts :~> c
apply' (HVecView ts
forall (xs :: [*]). KnownTypeList xs => HVecView xs
viewHVec' :: HVecView ts)

applyNAry' :: KnownTypeList ts => proxy ts -> proxy' c -> (HVec ts -> c) -> ts :~> c
applyNAry' :: proxy ts -> proxy' c -> (HVec ts -> c) -> ts :~> c
applyNAry' proxy ts
_ proxy' c
_ = (HVec ts -> c) -> ts :~> c
forall (ts :: [*]) c.
KnownTypeList ts =>
(HVec ts -> c) -> ts :~> c
applyNAry

class FromBool (c :: *) where
  type Predicate c :: Bool
  type Args c :: [*]
  fromBool :: Predicate c ~ 'True => HVec (Args c) -> c

fromBool' :: forall proxy c. (KnownTypeList (Args c), FromBool c , Predicate c ~ 'True) => proxy c -> Args c :~> c
fromBool' :: proxy c -> Args c :~> c
fromBool' proxy c
pxyc = Proxy (Args c) -> proxy c -> (HVec (Args c) -> c) -> Args c :~> c
forall (ts :: [*]) (proxy :: [*] -> *) (proxy' :: * -> *) c.
KnownTypeList ts =>
proxy ts -> proxy' c -> (HVec ts -> c) -> ts :~> c
applyNAry' (Proxy (Args c)
forall k (t :: k). Proxy t
Proxy :: Proxy (Args c)) proxy c
pxyc HVec (Args c) -> c
forall c. (FromBool c, Predicate c ~ 'True) => HVec (Args c) -> c
fromBool