{-# LANGUAGE ConstraintKinds        #-}
{-# LANGUAGE DataKinds              #-}
{-# LANGUAGE FlexibleContexts       #-}
{-# LANGUAGE FlexibleInstances      #-}
{-# LANGUAGE FunctionalDependencies #-}
{-# LANGUAGE GADTs                  #-}
{-# LANGUAGE KindSignatures         #-}
{-# LANGUAGE MultiParamTypeClasses  #-}
{-# LANGUAGE PolyKinds              #-}
{-# LANGUAGE RankNTypes             #-}
{-# LANGUAGE ScopedTypeVariables    #-}
{-# LANGUAGE TypeFamilies           #-}
{-# LANGUAGE TypeOperators          #-}
{-# LANGUAGE TypeSynonymInstances   #-}
{-# LANGUAGE UndecidableInstances   #-}

--------------------------------------------------------------------------------
-- |
-- Module      :  Data.Comp.Ops
-- Copyright   :  (c) 2011 Patrick Bahr
-- License     :  BSD3
-- Maintainer  :  Patrick Bahr <paba@diku.dk>
-- Stability   :  experimental
-- Portability :  non-portable (GHC Extensions)
--
-- This module provides operators on higher-order functors. All definitions are
-- generalised versions of those in "Data.Comp.Ops".
--
--------------------------------------------------------------------------------

module Data.Comp.Multi.Ops 
    ( module Data.Comp.Multi.Ops
    , (O.:*:)(..)
    , O.ffst
    , O.fsnd
    ) where


import Control.Monad
import Data.Kind

import Data.Comp.Multi.HFoldable
import Data.Comp.Multi.HFunctor
import Data.Comp.Multi.HTraversable
import qualified Data.Comp.Ops as O

import Data.Comp.SubsumeCommon

infixr 6 :+:


-- |Data type defining coproducts.
data (f :+: g) (h :: Type -> Type) e = Inl (f h e)
                                     | Inr (g h e)

{-| Utility function to case on a higher-order functor sum, without exposing the
  internal representation of sums. -}
caseH :: (f a b -> c) -> (g a b -> c) -> (f :+: g) a b -> c
caseH :: forall {k} (f :: (* -> *) -> k -> *) (a :: * -> *) (b :: k) c
       (g :: (* -> *) -> k -> *).
(f a b -> c) -> (g a b -> c) -> (:+:) f g a b -> c
caseH f a b -> c
f g a b -> c
g (:+:) f g a b
x = case (:+:) f g a b
x of
                Inl f a b
x -> f a b -> c
f f a b
x
                Inr g a b
x -> g a b -> c
g g a b
x

instance (HFunctor f, HFunctor g) => HFunctor (f :+: g) where
    hfmap :: forall (f :: * -> *) (g :: * -> *).
(f :-> g) -> (:+:) f g f :-> (:+:) f g g
hfmap f :-> g
f (Inl f f i
v) = forall {k} (f :: (* -> *) -> k -> *) (g :: (* -> *) -> k -> *)
       (h :: * -> *) (e :: k).
f h e -> (:+:) f g h e
Inl forall a b. (a -> b) -> a -> b
$ forall (h :: (* -> *) -> * -> *) (f :: * -> *) (g :: * -> *).
HFunctor h =>
(f :-> g) -> h f :-> h g
hfmap f :-> g
f f f i
v
    hfmap f :-> g
f (Inr g f i
v) = forall {k} (f :: (* -> *) -> k -> *) (g :: (* -> *) -> k -> *)
       (h :: * -> *) (e :: k).
g h e -> (:+:) f g h e
Inr forall a b. (a -> b) -> a -> b
$ forall (h :: (* -> *) -> * -> *) (f :: * -> *) (g :: * -> *).
HFunctor h =>
(f :-> g) -> h f :-> h g
hfmap f :-> g
f g f i
v

instance (HFoldable f, HFoldable g) => HFoldable (f :+: g) where
    hfold :: forall m. Monoid m => (:+:) f g (K m) :=> m
hfold (Inl f (K m) i
e) = forall (h :: (* -> *) -> * -> *) m.
(HFoldable h, Monoid m) =>
h (K m) :=> m
hfold f (K m) i
e
    hfold (Inr g (K m) i
e) = forall (h :: (* -> *) -> * -> *) m.
(HFoldable h, Monoid m) =>
h (K m) :=> m
hfold g (K m) i
e
    hfoldMap :: forall m (a :: * -> *). Monoid m => (a :=> m) -> (:+:) f g a :=> m
hfoldMap a :=> m
f (Inl f a i
e) = forall (h :: (* -> *) -> * -> *) m (a :: * -> *).
(HFoldable h, Monoid m) =>
(a :=> m) -> h a :=> m
hfoldMap a :=> m
f f a i
e
    hfoldMap a :=> m
f (Inr g a i
e) = forall (h :: (* -> *) -> * -> *) m (a :: * -> *).
(HFoldable h, Monoid m) =>
(a :=> m) -> h a :=> m
hfoldMap a :=> m
f g a i
e
    hfoldr :: forall (a :: * -> *) b. (a :=> (b -> b)) -> b -> (:+:) f g a :=> b
hfoldr a :=> (b -> b)
f b
b (Inl f a i
e) = forall (h :: (* -> *) -> * -> *) (a :: * -> *) b.
HFoldable h =>
(a :=> (b -> b)) -> b -> h a :=> b
hfoldr a :=> (b -> b)
f b
b f a i
e
    hfoldr a :=> (b -> b)
f b
b (Inr g a i
e) = forall (h :: (* -> *) -> * -> *) (a :: * -> *) b.
HFoldable h =>
(a :=> (b -> b)) -> b -> h a :=> b
hfoldr a :=> (b -> b)
f b
b g a i
e
    hfoldl :: forall b (a :: * -> *). (b -> a :=> b) -> b -> (:+:) f g a :=> b
hfoldl b -> a :=> b
f b
b (Inl f a i
e) = forall (h :: (* -> *) -> * -> *) b (a :: * -> *).
HFoldable h =>
(b -> a :=> b) -> b -> h a :=> b
hfoldl b -> a :=> b
f b
b f a i
e
    hfoldl b -> a :=> b
f b
b (Inr g a i
e) = forall (h :: (* -> *) -> * -> *) b (a :: * -> *).
HFoldable h =>
(b -> a :=> b) -> b -> h a :=> b
hfoldl b -> a :=> b
f b
b g a i
e

    hfoldr1 :: forall a. (a -> a -> a) -> (:+:) f g (K a) :=> a
hfoldr1 a -> a -> a
f (Inl f (K a) i
e) = forall (h :: (* -> *) -> * -> *) a.
HFoldable h =>
(a -> a -> a) -> h (K a) :=> a
hfoldr1 a -> a -> a
f f (K a) i
e
    hfoldr1 a -> a -> a
f (Inr g (K a) i
e) = forall (h :: (* -> *) -> * -> *) a.
HFoldable h =>
(a -> a -> a) -> h (K a) :=> a
hfoldr1 a -> a -> a
f g (K a) i
e
    hfoldl1 :: forall a. (a -> a -> a) -> (:+:) f g (K a) :=> a
hfoldl1 a -> a -> a
f (Inl f (K a) i
e) = forall (h :: (* -> *) -> * -> *) a.
HFoldable h =>
(a -> a -> a) -> h (K a) :=> a
hfoldl1 a -> a -> a
f f (K a) i
e
    hfoldl1 a -> a -> a
f (Inr g (K a) i
e) = forall (h :: (* -> *) -> * -> *) a.
HFoldable h =>
(a -> a -> a) -> h (K a) :=> a
hfoldl1 a -> a -> a
f g (K a) i
e

instance (HTraversable f, HTraversable g) => HTraversable (f :+: g) where
    htraverse :: forall (f :: * -> *) (a :: * -> *) (b :: * -> *).
Applicative f =>
NatM f a b -> NatM f ((:+:) f g a) ((:+:) f g b)
htraverse NatM f a b
f (Inl f a i
e) = forall {k} (f :: (* -> *) -> k -> *) (g :: (* -> *) -> k -> *)
       (h :: * -> *) (e :: k).
f h e -> (:+:) f g h e
Inl forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> forall (t :: (* -> *) -> * -> *) (f :: * -> *) (a :: * -> *)
       (b :: * -> *).
(HTraversable t, Applicative f) =>
NatM f a b -> NatM f (t a) (t b)
htraverse NatM f a b
f f a i
e
    htraverse NatM f a b
f (Inr g a i
e) = forall {k} (f :: (* -> *) -> k -> *) (g :: (* -> *) -> k -> *)
       (h :: * -> *) (e :: k).
g h e -> (:+:) f g h e
Inr forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> forall (t :: (* -> *) -> * -> *) (f :: * -> *) (a :: * -> *)
       (b :: * -> *).
(HTraversable t, Applicative f) =>
NatM f a b -> NatM f (t a) (t b)
htraverse NatM f a b
f g a i
e
    hmapM :: forall (m :: * -> *) (a :: * -> *) (b :: * -> *).
Monad m =>
NatM m a b -> NatM m ((:+:) f g a) ((:+:) f g b)
hmapM NatM m a b
f (Inl f a i
e) = forall {k} (f :: (* -> *) -> k -> *) (g :: (* -> *) -> k -> *)
       (h :: * -> *) (e :: k).
f h e -> (:+:) f g h e
Inl forall (m :: * -> *) a1 r. Monad m => (a1 -> r) -> m a1 -> m r
`liftM` forall (t :: (* -> *) -> * -> *) (m :: * -> *) (a :: * -> *)
       (b :: * -> *).
(HTraversable t, Monad m) =>
NatM m a b -> NatM m (t a) (t b)
hmapM NatM m a b
f f a i
e
    hmapM NatM m a b
f (Inr g a i
e) = forall {k} (f :: (* -> *) -> k -> *) (g :: (* -> *) -> k -> *)
       (h :: * -> *) (e :: k).
g h e -> (:+:) f g h e
Inr forall (m :: * -> *) a1 r. Monad m => (a1 -> r) -> m a1 -> m r
`liftM` forall (t :: (* -> *) -> * -> *) (m :: * -> *) (a :: * -> *)
       (b :: * -> *).
(HTraversable t, Monad m) =>
NatM m a b -> NatM m (t a) (t b)
hmapM NatM m a b
f g a i
e

-- The subsumption relation.

infixl 5 :<:
infixl 5 :=:

type family Elem (f :: (Type -> Type) -> Type -> Type)
                 (g :: (Type -> Type) -> Type -> Type) :: Emb where
    Elem f f = Found Here
    Elem (f1 :+: f2) g =  Sum' (Elem f1 g) (Elem f2 g)
    Elem f (g1 :+: g2) = Choose (Elem f g1) (Elem f g2)
    Elem f g = NotFound

class Subsume (e :: Emb) (f :: (Type -> Type) -> Type -> Type)
                         (g :: (Type -> Type) -> Type -> Type) where
  inj'  :: Proxy e -> f a :-> g a
  prj'  :: Proxy e -> NatM Maybe (g a) (f a)

instance Subsume (Found Here) f f where
    inj' :: forall (a :: * -> *). Proxy ('Found 'Here) -> f a :-> f a
inj' Proxy ('Found 'Here)
_ = forall a. a -> a
id

    prj' :: forall (a :: * -> *).
Proxy ('Found 'Here) -> NatM Maybe (f a) (f a)
prj' Proxy ('Found 'Here)
_ = forall a. a -> Maybe a
Just

instance Subsume (Found p) f g => Subsume (Found (Le p)) f (g :+: g') where
    inj' :: forall (a :: * -> *).
Proxy ('Found ('Le p)) -> f a :-> (:+:) g g' a
inj' Proxy ('Found ('Le p))
_ = forall {k} (f :: (* -> *) -> k -> *) (g :: (* -> *) -> k -> *)
       (h :: * -> *) (e :: k).
f h e -> (:+:) f g h e
Inl forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (e :: Emb) (f :: (* -> *) -> * -> *)
       (g :: (* -> *) -> * -> *) (a :: * -> *).
Subsume e f g =>
Proxy e -> f a :-> g a
inj' (forall {k} (a :: k). Proxy a
P :: Proxy (Found p))

    prj' :: forall (a :: * -> *).
Proxy ('Found ('Le p)) -> NatM Maybe ((:+:) g g' a) (f a)
prj' Proxy ('Found ('Le p))
_ (Inl g a i
x) = forall (e :: Emb) (f :: (* -> *) -> * -> *)
       (g :: (* -> *) -> * -> *) (a :: * -> *).
Subsume e f g =>
Proxy e -> NatM Maybe (g a) (f a)
prj' (forall {k} (a :: k). Proxy a
P :: Proxy (Found p)) g a i
x
    prj' Proxy ('Found ('Le p))
_ (:+:) g g' a i
_       = forall a. Maybe a
Nothing

instance Subsume (Found p) f g => Subsume (Found (Ri p)) f (g' :+: g) where
    inj' :: forall (a :: * -> *).
Proxy ('Found ('Ri p)) -> f a :-> (:+:) g' g a
inj' Proxy ('Found ('Ri p))
_ = forall {k} (f :: (* -> *) -> k -> *) (g :: (* -> *) -> k -> *)
       (h :: * -> *) (e :: k).
g h e -> (:+:) f g h e
Inr forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall (e :: Emb) (f :: (* -> *) -> * -> *)
       (g :: (* -> *) -> * -> *) (a :: * -> *).
Subsume e f g =>
Proxy e -> f a :-> g a
inj' (forall {k} (a :: k). Proxy a
P :: Proxy (Found p))

    prj' :: forall (a :: * -> *).
Proxy ('Found ('Ri p)) -> NatM Maybe ((:+:) g' g a) (f a)
prj' Proxy ('Found ('Ri p))
_ (Inr g a i
x) = forall (e :: Emb) (f :: (* -> *) -> * -> *)
       (g :: (* -> *) -> * -> *) (a :: * -> *).
Subsume e f g =>
Proxy e -> NatM Maybe (g a) (f a)
prj' (forall {k} (a :: k). Proxy a
P :: Proxy (Found p)) g a i
x
    prj' Proxy ('Found ('Ri p))
_ (:+:) g' g a i
_       = forall a. Maybe a
Nothing

instance (Subsume (Found p1) f1 g, Subsume (Found p2) f2 g)
    => Subsume (Found (Sum p1 p2)) (f1 :+: f2) g where
    inj' :: forall (a :: * -> *).
Proxy ('Found ('Sum p1 p2)) -> (:+:) f1 f2 a :-> g a
inj' Proxy ('Found ('Sum p1 p2))
_ (Inl f1 a i
x) = forall (e :: Emb) (f :: (* -> *) -> * -> *)
       (g :: (* -> *) -> * -> *) (a :: * -> *).
Subsume e f g =>
Proxy e -> f a :-> g a
inj' (forall {k} (a :: k). Proxy a
P :: Proxy (Found p1)) f1 a i
x
    inj' Proxy ('Found ('Sum p1 p2))
_ (Inr f2 a i
x) = forall (e :: Emb) (f :: (* -> *) -> * -> *)
       (g :: (* -> *) -> * -> *) (a :: * -> *).
Subsume e f g =>
Proxy e -> f a :-> g a
inj' (forall {k} (a :: k). Proxy a
P :: Proxy (Found p2)) f2 a i
x

    prj' :: forall (a :: * -> *).
Proxy ('Found ('Sum p1 p2)) -> NatM Maybe (g a) ((:+:) f1 f2 a)
prj' Proxy ('Found ('Sum p1 p2))
_ g a i
x = case forall (e :: Emb) (f :: (* -> *) -> * -> *)
       (g :: (* -> *) -> * -> *) (a :: * -> *).
Subsume e f g =>
Proxy e -> NatM Maybe (g a) (f a)
prj' (forall {k} (a :: k). Proxy a
P :: Proxy (Found p1)) g a i
x of
                 Just f1 a i
y -> forall a. a -> Maybe a
Just (forall {k} (f :: (* -> *) -> k -> *) (g :: (* -> *) -> k -> *)
       (h :: * -> *) (e :: k).
f h e -> (:+:) f g h e
Inl f1 a i
y)
                 Maybe (f1 a i)
_      -> case forall (e :: Emb) (f :: (* -> *) -> * -> *)
       (g :: (* -> *) -> * -> *) (a :: * -> *).
Subsume e f g =>
Proxy e -> NatM Maybe (g a) (f a)
prj' (forall {k} (a :: k). Proxy a
P :: Proxy (Found p2)) g a i
x of
                             Just f2 a i
y -> forall a. a -> Maybe a
Just (forall {k} (f :: (* -> *) -> k -> *) (g :: (* -> *) -> k -> *)
       (h :: * -> *) (e :: k).
g h e -> (:+:) f g h e
Inr f2 a i
y)
                             Maybe (f2 a i)
_      -> forall a. Maybe a
Nothing



-- | A constraint @f :<: g@ expresses that the signature @f@ is
-- subsumed by @g@, i.e. @f@ can be used to construct elements in @g@.
type f :<: g = (Subsume (ComprEmb (Elem f g)) f g)


inj :: forall f g a . (f :<: g) => f a :-> g a
inj :: forall (f :: (* -> *) -> * -> *) (g :: (* -> *) -> * -> *)
       (a :: * -> *).
(f :<: g) =>
f a :-> g a
inj = forall (e :: Emb) (f :: (* -> *) -> * -> *)
       (g :: (* -> *) -> * -> *) (a :: * -> *).
Subsume e f g =>
Proxy e -> f a :-> g a
inj' (forall {k} (a :: k). Proxy a
P :: Proxy (ComprEmb (Elem f g)))

proj :: forall f g a . (f :<: g) => NatM Maybe (g a) (f a)
proj :: forall (f :: (* -> *) -> * -> *) (g :: (* -> *) -> * -> *)
       (a :: * -> *).
(f :<: g) =>
NatM Maybe (g a) (f a)
proj = forall (e :: Emb) (f :: (* -> *) -> * -> *)
       (g :: (* -> *) -> * -> *) (a :: * -> *).
Subsume e f g =>
Proxy e -> NatM Maybe (g a) (f a)
prj' (forall {k} (a :: k). Proxy a
P :: Proxy (ComprEmb (Elem f g)))

type f :=: g = (f :<: g, g :<: f)



spl :: (f :=: f1 :+: f2) => (f1 a :-> b) -> (f2 a :-> b) -> f a :-> b
spl :: forall (f :: (* -> *) -> * -> *) (f1 :: (* -> *) -> * -> *)
       (f2 :: (* -> *) -> * -> *) (a :: * -> *) (b :: * -> *).
(f :=: (f1 :+: f2)) =>
(f1 a :-> b) -> (f2 a :-> b) -> f a :-> b
spl f1 a :-> b
f1 f2 a :-> b
f2 f a i
x = case forall (f :: (* -> *) -> * -> *) (g :: (* -> *) -> * -> *)
       (a :: * -> *).
(f :<: g) =>
f a :-> g a
inj f a i
x of
            Inl f1 a i
y -> f1 a :-> b
f1 f1 a i
y
            Inr f2 a i
y -> f2 a :-> b
f2 f2 a i
y

-- Constant Products

infixr 7 :&:

-- | This data type adds a constant product to a
-- signature. Alternatively, this could have also been defined as
--
-- @
-- data (f :&: a) (g ::  Type -> Type) e = f g e :&: a e
-- @
--
-- This is too general, however, for example for 'productHHom'.

data (f :&: a) (g ::  Type -> Type) e = f g e :&: a


instance (HFunctor f) => HFunctor (f :&: a) where
    hfmap :: forall (f :: * -> *) (g :: * -> *).
(f :-> g) -> (:&:) f a f :-> (:&:) f a g
hfmap f :-> g
f (f f i
v :&: a
c) = forall (h :: (* -> *) -> * -> *) (f :: * -> *) (g :: * -> *).
HFunctor h =>
(f :-> g) -> h f :-> h g
hfmap f :-> g
f f f i
v forall {k} (f :: (* -> *) -> k -> *) a (g :: * -> *) (e :: k).
f g e -> a -> (:&:) f a g e
:&: a
c

instance (HFoldable f) => HFoldable (f :&: a) where
    hfold :: forall m. Monoid m => (:&:) f a (K m) :=> m
hfold (f (K m) i
v :&: a
_) = forall (h :: (* -> *) -> * -> *) m.
(HFoldable h, Monoid m) =>
h (K m) :=> m
hfold f (K m) i
v
    hfoldMap :: forall m (a :: * -> *). Monoid m => (a :=> m) -> (:&:) f a a :=> m
hfoldMap a :=> m
f (f a i
v :&: a
_) = forall (h :: (* -> *) -> * -> *) m (a :: * -> *).
(HFoldable h, Monoid m) =>
(a :=> m) -> h a :=> m
hfoldMap a :=> m
f f a i
v
    hfoldr :: forall (a :: * -> *) b. (a :=> (b -> b)) -> b -> (:&:) f a a :=> b
hfoldr a :=> (b -> b)
f b
e (f a i
v :&: a
_) = forall (h :: (* -> *) -> * -> *) (a :: * -> *) b.
HFoldable h =>
(a :=> (b -> b)) -> b -> h a :=> b
hfoldr a :=> (b -> b)
f b
e f a i
v
    hfoldl :: forall b (a :: * -> *). (b -> a :=> b) -> b -> (:&:) f a a :=> b
hfoldl b -> a :=> b
f b
e (f a i
v :&: a
_) = forall (h :: (* -> *) -> * -> *) b (a :: * -> *).
HFoldable h =>
(b -> a :=> b) -> b -> h a :=> b
hfoldl b -> a :=> b
f b
e f a i
v
    hfoldr1 :: forall a. (a -> a -> a) -> (:&:) f a (K a) :=> a
hfoldr1 a -> a -> a
f (f (K a) i
v :&: a
_) = forall (h :: (* -> *) -> * -> *) a.
HFoldable h =>
(a -> a -> a) -> h (K a) :=> a
hfoldr1 a -> a -> a
f f (K a) i
v
    hfoldl1 :: forall a. (a -> a -> a) -> (:&:) f a (K a) :=> a
hfoldl1 a -> a -> a
f (f (K a) i
v :&: a
_) = forall (h :: (* -> *) -> * -> *) a.
HFoldable h =>
(a -> a -> a) -> h (K a) :=> a
hfoldl1 a -> a -> a
f f (K a) i
v


instance (HTraversable f) => HTraversable (f :&: a) where
    htraverse :: forall (f :: * -> *) (a :: * -> *) (b :: * -> *).
Applicative f =>
NatM f a b -> NatM f ((:&:) f a a) ((:&:) f a b)
htraverse NatM f a b
f (f a i
v :&: a
c) =  (forall {k} (f :: (* -> *) -> k -> *) a (g :: * -> *) (e :: k).
f g e -> a -> (:&:) f a g e
:&: a
c) forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> (forall (t :: (* -> *) -> * -> *) (f :: * -> *) (a :: * -> *)
       (b :: * -> *).
(HTraversable t, Applicative f) =>
NatM f a b -> NatM f (t a) (t b)
htraverse NatM f a b
f f a i
v)
    hmapM :: forall (m :: * -> *) (a :: * -> *) (b :: * -> *).
Monad m =>
NatM m a b -> NatM m ((:&:) f a a) ((:&:) f a b)
hmapM NatM m a b
f (f a i
v :&: a
c) = forall (m :: * -> *) a1 r. Monad m => (a1 -> r) -> m a1 -> m r
liftM (forall {k} (f :: (* -> *) -> k -> *) a (g :: * -> *) (e :: k).
f g e -> a -> (:&:) f a g e
:&: a
c) (forall (t :: (* -> *) -> * -> *) (m :: * -> *) (a :: * -> *)
       (b :: * -> *).
(HTraversable t, Monad m) =>
NatM m a b -> NatM m (t a) (t b)
hmapM NatM m a b
f f a i
v)

-- | This class defines how to distribute an annotation over a sum of
-- signatures.
class DistAnn (s :: (Type -> Type) -> Type -> Type) p s' | s' -> s, s' -> p where
    -- | This function injects an annotation over a signature.
    injectA :: p -> s a :-> s' a
    projectA :: s' a :-> (s a O.:&: p)


class RemA (s :: (Type -> Type) -> Type -> Type) s' | s -> s'  where
    remA :: s a :-> s' a


instance (RemA s s') => RemA (f :&: p :+: s) (f :+: s') where
    remA :: forall (a :: * -> *). (:+:) (f :&: p) s a :-> (:+:) f s' a
remA (Inl (f a i
v :&: p
_)) = forall {k} (f :: (* -> *) -> k -> *) (g :: (* -> *) -> k -> *)
       (h :: * -> *) (e :: k).
f h e -> (:+:) f g h e
Inl f a i
v
    remA (Inr s a i
v) = forall {k} (f :: (* -> *) -> k -> *) (g :: (* -> *) -> k -> *)
       (h :: * -> *) (e :: k).
g h e -> (:+:) f g h e
Inr forall a b. (a -> b) -> a -> b
$ forall (s :: (* -> *) -> * -> *) (s' :: (* -> *) -> * -> *)
       (a :: * -> *).
RemA s s' =>
s a :-> s' a
remA s a i
v


instance RemA (f :&: p) f where
    remA :: forall (a :: * -> *). (:&:) f p a :-> f a
remA (f a i
v :&: p
_) = f a i
v


instance DistAnn f p (f :&: p) where

    injectA :: forall (a :: * -> *). p -> f a :-> (:&:) f p a
injectA p
p f a i
v = f a i
v forall {k} (f :: (* -> *) -> k -> *) a (g :: * -> *) (e :: k).
f g e -> a -> (:&:) f a g e
:&: p
p

    projectA :: forall (a :: * -> *). (:&:) f p a :-> (f a :&: p)
projectA (f a i
v :&: p
p) = f a i
v forall {k} (f :: k -> *) a (e :: k). f e -> a -> (:&:) f a e
O.:&: p
p


instance (DistAnn s p s') => DistAnn (f :+: s) p ((f :&: p) :+: s') where
    injectA :: forall (a :: * -> *). p -> (:+:) f s a :-> (:+:) (f :&: p) s' a
injectA p
p (Inl f a i
v) = forall {k} (f :: (* -> *) -> k -> *) (g :: (* -> *) -> k -> *)
       (h :: * -> *) (e :: k).
f h e -> (:+:) f g h e
Inl (f a i
v forall {k} (f :: (* -> *) -> k -> *) a (g :: * -> *) (e :: k).
f g e -> a -> (:&:) f a g e
:&: p
p)
    injectA p
p (Inr s a i
v) = forall {k} (f :: (* -> *) -> k -> *) (g :: (* -> *) -> k -> *)
       (h :: * -> *) (e :: k).
g h e -> (:+:) f g h e
Inr forall a b. (a -> b) -> a -> b
$ forall (s :: (* -> *) -> * -> *) p (s' :: (* -> *) -> * -> *)
       (a :: * -> *).
DistAnn s p s' =>
p -> s a :-> s' a
injectA p
p s a i
v

    projectA :: forall (a :: * -> *). (:+:) (f :&: p) s' a :-> ((:+:) f s a :&: p)
projectA (Inl (f a i
v :&: p
p)) = (forall {k} (f :: (* -> *) -> k -> *) (g :: (* -> *) -> k -> *)
       (h :: * -> *) (e :: k).
f h e -> (:+:) f g h e
Inl f a i
v forall {k} (f :: k -> *) a (e :: k). f e -> a -> (:&:) f a e
O.:&: p
p)
    projectA (Inr s' a i
v) = let (s a i
v' O.:&: p
p) = forall (s :: (* -> *) -> * -> *) p (s' :: (* -> *) -> * -> *)
       (a :: * -> *).
DistAnn s p s' =>
s' a :-> (s a :&: p)
projectA s' a i
v
                        in  (forall {k} (f :: (* -> *) -> k -> *) (g :: (* -> *) -> k -> *)
       (h :: * -> *) (e :: k).
g h e -> (:+:) f g h e
Inr s a i
v' forall {k} (f :: k -> *) a (e :: k). f e -> a -> (:&:) f a e
O.:&: p
p)