# LANGUAGE 
   RankNTypes , 
   NoImplicitPrelude , 
   GADTs , 
   ExplicitForAll , 
   ScopedTypeVariables , 
   NoMonomorphismRestriction , 
   IncoherentInstances , 
   InstanceSigs , 
   LambdaCase , 
   FlexibleContexts , 
   KindSignatures , 
   TypeFamilies , 
   TypeApplications , 
   MultiParamTypeClasses , 
   FlexibleInstances , 
   TypeOperators 
 #
  RankNTypes,
  NoImplicitPrelude,
  GADTs,
  ExplicitForAll,
  ScopedTypeVariables,
  NoMonomorphismRestriction,
  IncoherentInstances,
  InstanceSigs,
  LambdaCase,
  FlexibleContexts,
  KindSignatures,
  TypeFamilies,
  TypeApplications,
  MultiParamTypeClasses,
  FlexibleInstances,
  TypeOperators
#-}

module DDF.PE where

import DDF.Lang
import qualified Prelude as M
import qualified DDF.Meta.Dual as M

type instance OrdC (P repr) = Ord repr

data P repr h a where
  Open   :: (forall hout. EnvT repr h hout -> P repr hout a) -> P repr h a
  Unk    :: repr h a -> P repr h a
  Known  ::
    K repr h a ->
    repr h a ->
    (forall hout. EnvT repr h hout -> P repr hout a) ->
    (forall any. P repr (any, h) a) ->
    (forall hh ht. (hh, ht) ~ h => P repr ht (hh -> a)) ->
    P repr h a

know :: DBI repr =>
  K repr h a ->
  repr h a ->
  (forall hout. EnvT repr h hout -> P repr hout a) ->
  (forall any. P repr (any, h) a) ->
  P repr h a
know a b c d = Known a b c d (mkFun c)

static :: forall repr h a. DBI repr => (forall h'. (K repr h' a, repr h' a)) -> P repr h a
static x = know (M.fst $ x @h) (M.snd x) (\_ -> static x) (static x)

isOpen (Open _) = M.True
isOpen _ = M.False

type family K (repr :: * -> * -> *) h a

mkFun :: DBI repr => (forall hout. EnvT repr (a, h) hout -> P repr hout b) -> P repr h (a -> b)
mkFun f = Known (Fun f) (abs $ dynamic (f Dyn)) (\h -> abs $ f $ Next h) (abs $ f $ Next Weak) (mkFun $ app_open (mkFun f))

data EnvT repr hin hout where
  Dyn  :: EnvT repr hin hin
  Arg  :: P repr hout a -> EnvT repr (a, hout) hout
  Weak :: EnvT repr h (a, h)
  Next :: EnvT repr hin hout -> EnvT repr (a, hin) (a, hout)

dynamic:: DBI repr => P repr h a -> repr h a
dynamic (Unk x)      = x
dynamic (Open f)     = dynamic (f Dyn)
dynamic (Known _ d _ _ _)  = d

app_open :: DBI repr => P repr hin r -> EnvT repr hin hout -> P repr hout r
app_open (Open fs) h       = fs h
app_open (Unk e) Dyn       = Unk e
app_open (Unk e) (Arg p)   = Unk (app (abs e) (dynamic p))
app_open (Unk e) (Next h)  = app (s (app_open (Unk (abs e)) h)) z
app_open (Unk e) Weak      = Unk (s e)
app_open (Known _ _ x _ _) h = x h

type instance K repr h (a -> b) = Fun repr h a b
newtype Fun repr h a b = Fun {runFun :: forall hout. EnvT repr (a, h) hout -> P repr hout b}

instance DBI r => DBI (P r) where
  z = Open f where
    f :: EnvT r (a,h) hout -> P r hout a
    f Dyn      = Unk z
    f (Arg x)  = x
    f (Next _) = z
    f Weak     = s z

  s :: forall h a any. P r h a -> P r (any, h) a
  s (Unk x) = Unk (s x)
  s (Known _ _ _ x _) = x
  s p@(Open _) = Open f where
    f :: EnvT r (any, h) hout -> P r hout a
    f Dyn              = Unk (s (dynamic p))
    f (Arg _)          = p
    f (Next h)         = s (app_open p h)
    f Weak             = s (s p)

  abs (Unk f) = Unk (abs f)
  abs (Open o) = mkFun o
  abs (Known _ _ _ _ x) = x

  app (Known (Fun fs) _ _ _ _) p     = fs (Arg p)
  app e1 e2 | isOpen e1 || isOpen e2 = Open (\h -> app (app_open e1 h) (app_open e2 h))
  app f x                            = Unk (app (dynamic f) (dynamic x))

type instance K repr h M.Bool = M.Bool
instance Bool r => Bool (P r) where
  bool x = static (x, bool x)
  ite = lam3 (\l r b -> app2 (f b) l r)
    where
      f :: P r h M.Bool -> P r h (a -> a -> a)
      f (Known M.True _ _ _ _) = const
      f (Known M.False _ _ _ _) = const1 id
      f (Open x) = Open $ f . x
      f (Unk x) = Unk (lam2 (\l r -> ite3 l r (s (s x))))

type instance K repr h M.Double = M.Double
instance Double r => Double (P r) where
  double x = static (x, double x)
  doublePlus = abs (abs (f (s z) z))
    where
      f :: P r h M.Double -> P r h M.Double -> P r h M.Double
      f (Known l _ _ _ _) (Known r _ _ _ _) = double (l + r)
      f (Known 0 _ _ _ _) r = r
      f l (Known 0 _ _ _ _) = l
      f l r | isOpen l || isOpen r = Open (\h -> f (app_open l h) (app_open r h))
      f l r = Unk (doublePlus2 (dynamic l) (dynamic r))
  doubleMult = abs (abs (f (s z) z))
    where
      f :: P r h M.Double -> P r h M.Double -> P r h M.Double
      f (Known l _ _ _ _) (Known r _ _ _ _) = double (l * r)
      f (Known 0 _ _ _ _) _ = double 0
      f _ (Known 0 _ _ _ _) = double 0
      f l (Known 1 _ _ _ _) = l
      f (Known 1 _ _ _ _) r = r
      f l r | isOpen l || isOpen r = Open (\h -> f (app_open l h) (app_open r h))
      f l r = Unk (doubleMult2 (dynamic l) (dynamic r))
  doubleMinus = abs (abs (f (s z) z))
    where
      f :: P r h M.Double -> P r h M.Double -> P r h M.Double
      f (Known l _ _ _ _) (Known r _ _ _ _) = double (l - r)
      f l (Known 0 _ _ _ _) = l 
      f l r | isOpen l || isOpen r = Open (\h -> f (app_open l h) (app_open r h))
      f l r = Unk $ doubleMinus2 (dynamic l) (dynamic r)
  doubleDivide = abs (abs (f (s z) z))
    where
      f :: P r h M.Double -> P r h M.Double -> P r h M.Double
      f (Known l _ _ _ _) (Known r _ _ _ _) = double (l / r)
      f (Known 0 _ _ _ _) _ = double 0
      f l (Known 1 _ _ _ _) = l 
      f l r | isOpen l || isOpen r = Open (\h -> f (app_open l h) (app_open r h))
      f l r = Unk $ doubleDivide2 (dynamic l) (dynamic r)
  doubleExp = abs (f z)
    where
      f :: P r h M.Double -> P r h M.Double
      f (Known l _ _ _ _) = double (M.exp l) 
      f (Open x) = Open $ f . x
      f (Unk l) = Unk $ doubleExp1 l
  doubleCmp = abs $ abs $ f (s z) z where
    f :: P r h M.Double -> P r h M.Double -> P r h M.Ordering
    f (Known l _ _ _ _) (Known r _ _ _ _) = ordering (M.compare l r)
    f l r | isOpen l || isOpen r = Open (\h -> f (app_open l h) (app_open r h))
    f l r = Unk (cmp2 (dynamic l) (dynamic r))

type instance K repr h M.Float = M.Float
instance Float r => Float (P r) where
  float x = static (x, float x)
  floatPlus = abs (abs (f (s z) z))
    where
      f :: P r h M.Float -> P r h M.Float -> P r h M.Float
      f (Known l _ _ _ _) (Known r _ _ _ _) = float (l + r)
      f (Known 0 _ _ _ _) r = r
      f l (Known 0 _ _ _ _) = l
      f l r | isOpen l || isOpen r = Open (\h -> f (app_open l h) (app_open r h))
      f l r = Unk (floatPlus2 (dynamic l) (dynamic r))
  floatMult = abs (abs (f (s z) z))
    where
      f :: P r h M.Float -> P r h M.Float -> P r h M.Float
      f (Known l _ _ _ _) (Known r _ _ _ _) = float (l * r)
      f (Known 0 _ _ _ _) _ = float 0
      f _ (Known 0 _ _ _ _) = float 0
      f l (Known 1 _ _ _ _) = l
      f (Known 1 _ _ _ _) r = r
      f l r | isOpen l || isOpen r = Open (\h -> f (app_open l h) (app_open r h))
      f l r = Unk (floatMult2 (dynamic l) (dynamic r))
  floatMinus = abs (abs (f (s z) z))
    where
      f :: P r h M.Float -> P r h M.Float -> P r h M.Float
      f (Known l _ _ _ _) (Known r _ _ _ _) = float (l - r)
      f l (Known 0 _ _ _ _) = l 
      f l r | isOpen l || isOpen r = Open (\h -> f (app_open l h) (app_open r h))
      f l r = Unk (floatMinus2 (dynamic l) (dynamic r))
  floatDivide = abs (abs (f (s z) z))
    where
      f :: P r h M.Float -> P r h M.Float -> P r h M.Float
      f (Known l _ _ _ _) (Known r _ _ _ _) = float (l / r)
      f (Known 0 _ _ _ _) _ = float 0
      f l (Known 1 _ _ _ _) = l 
      f l r | isOpen l || isOpen r = Open (\h -> f (app_open l h) (app_open r h))
      f l r = Unk (floatDivide2 (dynamic l) (dynamic r))
  floatExp = abs (f z)
    where
      f :: P r h M.Float -> P r h M.Float
      f (Known l _ _ _ _) = float (M.exp l) 
      f (Open x) = Open $ f . x
      f (Unk l) = Unk (floatExp1 l)

type instance K repr h (a, b) = (P repr h a, P repr h b)
instance Prod r => Prod (P r) where
  mkProd = abs (abs (f (s z) z))
    where
      f :: P r h a -> P r h b -> P r h (a, b)
      f l r = know (l, r)
                (mkProd2 (dynamic l) (dynamic r))
                (\h -> mkProd2 (app_open l h) (app_open r h))
                (mkProd2 (s l) (s r))
  zro = abs (f z)
    where
      f :: P r h (a, b) -> P r h a
      f (Known (l, _) _ _ _ _) = l
      f (Open x) = Open $ f . x
      f (Unk p) = Unk (zro1 p)
  fst = abs (f z)
    where
      f :: P r h (a, b) -> P r h b
      f (Known (_, r) _ _ _ _) = r
      f (Open x) = Open $ f . x
      f (Unk p) = Unk (fst1 p)

type instance K repr h (M.Either a b) = M.Either (P repr h a) (P repr h b)
instance Sum r => Sum (P r) where
  left = abs (f z)
    where
      f :: P r h a -> P r h (M.Either a b)
      f x = know (Left x)
              (left1 $ dynamic x)
              (\h -> left1 $ app_open x h)
              (left1 $ s x)
  right = abs (f z)
    where
      f :: P r h b -> P r h (M.Either a b)
      f x = know (Right x)
              (right1 $ dynamic x)
              (\h -> right1 $ app_open x h)
              (right1 $ s x)
  sumMatch = abs $ abs $ abs (f (s (s z)) (s z) z)
    where
      f :: P r h (a -> c) -> P r h (b -> c) -> P r h (M.Either a b) -> P r h c
      f l _ (Known (M.Left x) _ _ _ _) = app l x
      f _ r (Known (M.Right x) _ _ _ _) = app r x
      f l r (Open x) = Open $ \h -> f (app_open l h) (app_open r h) (x h)
      f l r (Unk x) = Unk $ sumMatch3 (dynamic l) (dynamic r) x

instance Y r => Y (P r) where
  y = Unk y -- naive strategy to avoid infinite loop in PE. Later might do infinite PE thx to laziness.

type instance K repr h [a] = Maybe (P repr h a, P repr h [a])
instance List repr => List (P repr) where
  nil = static (Nothing, nil)
  cons = abs $ abs (f (s z) z)
    where
      f :: P repr h a -> P repr h [a] -> P repr h [a]
      f h t = know (Just (h, t))
                (cons2 (dynamic h) (dynamic t))
                (\env -> cons2 (app_open h env) (app_open t env))
                (cons2 (s h) (s t))
  listMatch = abs $ abs $ abs (f (s $ s z) (s z) z)
    where
      f :: P repr h b -> P repr h (a -> [a] -> b) -> P repr h [a] -> P repr h b
  You know nothing ,   .
      f _ r (Known (Just (h, t)) _ _ _ _) = app2 r h t
      f l r (Open x) = Open $ \h -> f (app_open l h) (app_open r h) (x h)
      f l r (Unk x) = Unk $ listMatch3 (dynamic l) (dynamic r) x
  listAppend = abs $ abs (f (s z) z)
    where
      f :: P repr h [a] -> P repr h [a] -> P repr h [a]
      f (Known Nothing _ _ _ _) r = r
      f (Known (Just (h, t)) _ _ _ _) r = cons2 h (listAppend2 t r)
      f l (Known Nothing _ _ _ _) = l
      f l r | isOpen l || isOpen r = Open $ \h -> f (app_open l h) (app_open r h)
      f l r = Unk (listAppend2 (dynamic l) (dynamic r))

type instance K repr h (Maybe a) = Maybe (P repr h a)
instance Option repr => Option (P repr) where
  nothing = static (Nothing, nothing)
  just = abs (f z)
    where
      f :: P repr h a -> P repr h (Maybe a)
      f x = know (Just x)
              (just1 $ dynamic x)
              (\h -> just1 $ app_open x h)
              (just1 $ s x)
  optionMatch = abs $ abs $ abs (f (s (s z)) (s z) z)
    where
      f :: P repr h b -> P repr h (a -> b) -> P repr h (Maybe a) -> P repr h b
      f l _ (Known Nothing _ _ _ _) = l
      f _ r (Known (Just x) _ _ _ _) = app r x
      f l r (Open x) = Open $ \h -> f (app_open l h) (app_open r h) (x h)
      f l r (Unk x) = Unk $ optionMatch3 (dynamic l) (dynamic r) x

type instance K repr h M.Char = M.Char
instance Char repr => Char (P repr) where
  char x = static (x, char x)

type instance K repr h M.Ordering = M.Ordering
instance Ordering repr => Ordering (P repr) where
  ordering x = static (x, ordering x)
  sel = abs $ abs $ abs $ abs $ f (s (s (s z))) (s (s z)) (s z) z where
    f :: P repr h a -> P repr h a -> P repr h a -> P repr h M.Ordering -> P repr h a
    f a _ _ (Known M.LT _ _ _ _) = a
    f _ b _ (Known M.EQ _ _ _ _) = b
    f _ _ c (Known M.GT _ _ _ _) = c
    f a b c (Open x) = Open $ \h -> f (app_open a h) (app_open b h) (app_open c h) (x h)
    f a b c (Unk x) = Unk $ sel4 (dynamic a) (dynamic b) (dynamic c) x

type instance K repr h M.Int = M.Int
instance Int repr => Int (P repr) where
  int x = static (x, int x)
  pred = abs (f z) where
    f :: P repr h M.Int -> P repr h M.Int
    f (Known i _ _ _ _) = int $ i - 1
    f (Open x) = Open $ f . x
    f (Unk x) = Unk $ pred1 x
  intCmp = abs $ abs $ f (s z) z where
    f :: P repr h M.Int -> P repr h M.Int -> P repr h M.Ordering
    f (Known l _ _ _ _) (Known r _ _ _ _) = ordering $ M.compare l r
    f l r | isOpen l || isOpen r = Open $ \h -> f (app_open l h) (app_open r h)
    f l r = Unk $ cmp2 (dynamic l) (dynamic r)

type instance K repr h (M.Dual l r) = (P repr h l, P repr h r)
instance Dual repr => Dual (P repr) where
  dual = abs $ f z
    where
      f :: P repr h (a, b) -> P repr h (M.Dual a b)
      f (Known (l, r) _ _ _ _) =
        know (l, r)
          (mkDual2 (dynamic l) (dynamic r))
          (\h -> mkDual2 (app_open l h) (app_open r h))
          (s $ mkDual2 l r)
      f (Open x) = Open $ f . x
      f (Unk x) = Unk $ dual1 x
  runDual = abs (f z)
    where
      f :: P repr h (M.Dual a b) -> P repr h (a, b)
      f (Known (l, r) _ _ _ _) =
        know (l, r)
          (mkProd2 (dynamic l) (dynamic r))
          (\h -> mkProd2 (app_open l h) (app_open r h))
          (mkProd2 (s l) (s r))
      f (Open x) = Open $ f . x
      f (Unk x) = Unk $ runDual1 x
  dualGetOrdC :: forall x y. Ord (P repr) x :- OrdC (P repr) (M.Dual x y)
  dualGetOrdC = Sub (withDict (getOrdC @(P repr) @x Proxy) Dict)

type instance K r h () = ()
instance Unit r => Unit (P r) where
  unit = static ((), unit)

type instance K r h (M.IO a) = P r h a
instance IO r => IO (P r) where
  putStrLn = Unk putStrLn
  ioJoin = abs $ f z where
    f :: P r h (M.IO (M.IO a)) -> P r h (M.IO a)
    f (Known l _ _ _ _) = l
    f (Open x) = Open $ f . x
    f (Unk x) = Unk $ join1 x
  ioMap = abs $ abs $ f (s z) z where
    f :: P r h (a -> b) -> P r h (M.IO a) -> P r h (M.IO b)
    f l (Known a _ _ _ _) = pure1 $ app l a
    f l (Open x) = Open $ \h -> f (app_open l h) (x h)
    f l (Unk x) = Unk $ map2 (dynamic l) x
  ioPure = abs $ f z where
    f :: P r h a -> P r h (M.IO a)
    f x = know x (pure1 $ dynamic x) (\h -> pure1 $ app_open x h) (pure1 $ s x)
  ioAP = abs $ abs $ f (s z) z where
    f :: P r h (M.IO (a -> b)) -> P r h (M.IO a) -> P r h (M.IO b)
    f (Known l _ _ _ _) (Known r _ _ _ _) = pure1 $ app l r
    f l r | isOpen l || isOpen r = Open $ \h -> f (app_open l h) (app_open r h)
    f l r = Unk $ ap2 (dynamic l) (dynamic r)

pe :: DBI repr => P repr () a -> repr () a
pe = dynamic

 naive strategy to avoid infinite loop in PE. Later might do infinite PE thx to laziness.
# LANGUAGE 
   RankNTypes , 
   NoImplicitPrelude , 
   GADTs , 
   ExplicitForAll , 
   ScopedTypeVariables , 
   NoMonomorphismRestriction , 
   IncoherentInstances , 
   InstanceSigs , 
   LambdaCase , 
   FlexibleContexts , 
   KindSignatures , 
   TypeFamilies , 
   TypeApplications , 
   MultiParamTypeClasses , 
   FlexibleInstances , 
   TypeOperators 
 #
  RankNTypes,
  NoImplicitPrelude,
  GADTs,
  ExplicitForAll,
  ScopedTypeVariables,
  NoMonomorphismRestriction,
  IncoherentInstances,
  InstanceSigs,
  LambdaCase,
  FlexibleContexts,
  KindSignatures,
  TypeFamilies,
  TypeApplications,
  MultiParamTypeClasses,
  FlexibleInstances,
  TypeOperators
#-}

module DDF.PE where

import DDF.Lang
import qualified Prelude as M
import qualified DDF.Meta.Dual as M

type instance OrdC (P repr) = Ord repr

data P repr h a where
  Open   :: (forall hout. EnvT repr h hout -> P repr hout a) -> P repr h a
  Unk    :: repr h a -> P repr h a
  Known  ::
    K repr h a ->
    repr h a ->
    (forall hout. EnvT repr h hout -> P repr hout a) ->
    (forall any. P repr (any, h) a) ->
    (forall hh ht. (hh, ht) ~ h => P repr ht (hh -> a)) ->
    P repr h a

know :: DBI repr =>
  K repr h a ->
  repr h a ->
  (forall hout. EnvT repr h hout -> P repr hout a) ->
  (forall any. P repr (any, h) a) ->
  P repr h a
know a b c d = Known a b c d (mkFun c)

static :: forall repr h a. DBI repr => (forall h'. (K repr h' a, repr h' a)) -> P repr h a
static x = know (M.fst $ x @h) (M.snd x) (\_ -> static x) (static x)

isOpen (Open _) = M.True
isOpen _ = M.False

type family K (repr :: * -> * -> *) h a

mkFun :: DBI repr => (forall hout. EnvT repr (a, h) hout -> P repr hout b) -> P repr h (a -> b)
mkFun f = Known (Fun f) (abs $ dynamic (f Dyn)) (\h -> abs $ f $ Next h) (abs $ f $ Next Weak) (mkFun $ app_open (mkFun f))

data EnvT repr hin hout where
  Dyn  :: EnvT repr hin hin
  Arg  :: P repr hout a -> EnvT repr (a, hout) hout
  Weak :: EnvT repr h (a, h)
  Next :: EnvT repr hin hout -> EnvT repr (a, hin) (a, hout)

dynamic:: DBI repr => P repr h a -> repr h a
dynamic (Unk x)      = x
dynamic (Open f)     = dynamic (f Dyn)
dynamic (Known _ d _ _ _)  = d

app_open :: DBI repr => P repr hin r -> EnvT repr hin hout -> P repr hout r
app_open (Open fs) h       = fs h
app_open (Unk e) Dyn       = Unk e
app_open (Unk e) (Arg p)   = Unk (app (abs e) (dynamic p))
app_open (Unk e) (Next h)  = app (s (app_open (Unk (abs e)) h)) z
app_open (Unk e) Weak      = Unk (s e)
app_open (Known _ _ x _ _) h = x h

type instance K repr h (a -> b) = Fun repr h a b
newtype Fun repr h a b = Fun {runFun :: forall hout. EnvT repr (a, h) hout -> P repr hout b}

instance DBI r => DBI (P r) where
  z = Open f where
    f :: EnvT r (a,h) hout -> P r hout a
    f Dyn      = Unk z
    f (Arg x)  = x
    f (Next _) = z
    f Weak     = s z

  s :: forall h a any. P r h a -> P r (any, h) a
  s (Unk x) = Unk (s x)
  s (Known _ _ _ x _) = x
  s p@(Open _) = Open f where
    f :: EnvT r (any, h) hout -> P r hout a
    f Dyn              = Unk (s (dynamic p))
    f (Arg _)          = p
    f (Next h)         = s (app_open p h)
    f Weak             = s (s p)

  abs (Unk f) = Unk (abs f)
  abs (Open o) = mkFun o
  abs (Known _ _ _ _ x) = x

  app (Known (Fun fs) _ _ _ _) p     = fs (Arg p)
  app e1 e2 | isOpen e1 || isOpen e2 = Open (\h -> app (app_open e1 h) (app_open e2 h))
  app f x                            = Unk (app (dynamic f) (dynamic x))

type instance K repr h M.Bool = M.Bool
instance Bool r => Bool (P r) where
  bool x = static (x, bool x)
  ite = lam3 (\l r b -> app2 (f b) l r)
    where
      f :: P r h M.Bool -> P r h (a -> a -> a)
      f (Known M.True _ _ _ _) = const
      f (Known M.False _ _ _ _) = const1 id
      f (Open x) = Open $ f . x
      f (Unk x) = Unk (lam2 (\l r -> ite3 l r (s (s x))))

type instance K repr h M.Double = M.Double
instance Double r => Double (P r) where
  double x = static (x, double x)
  doublePlus = abs (abs (f (s z) z))
    where
      f :: P r h M.Double -> P r h M.Double -> P r h M.Double
      f (Known l _ _ _ _) (Known r _ _ _ _) = double (l + r)
      f (Known 0 _ _ _ _) r = r
      f l (Known 0 _ _ _ _) = l
      f l r | isOpen l || isOpen r = Open (\h -> f (app_open l h) (app_open r h))
      f l r = Unk (doublePlus2 (dynamic l) (dynamic r))
  doubleMult = abs (abs (f (s z) z))
    where
      f :: P r h M.Double -> P r h M.Double -> P r h M.Double
      f (Known l _ _ _ _) (Known r _ _ _ _) = double (l * r)
      f (Known 0 _ _ _ _) _ = double 0
      f _ (Known 0 _ _ _ _) = double 0
      f l (Known 1 _ _ _ _) = l
      f (Known 1 _ _ _ _) r = r
      f l r | isOpen l || isOpen r = Open (\h -> f (app_open l h) (app_open r h))
      f l r = Unk (doubleMult2 (dynamic l) (dynamic r))
  doubleMinus = abs (abs (f (s z) z))
    where
      f :: P r h M.Double -> P r h M.Double -> P r h M.Double
      f (Known l _ _ _ _) (Known r _ _ _ _) = double (l - r)
      f l (Known 0 _ _ _ _) = l 
      f l r | isOpen l || isOpen r = Open (\h -> f (app_open l h) (app_open r h))
      f l r = Unk $ doubleMinus2 (dynamic l) (dynamic r)
  doubleDivide = abs (abs (f (s z) z))
    where
      f :: P r h M.Double -> P r h M.Double -> P r h M.Double
      f (Known l _ _ _ _) (Known r _ _ _ _) = double (l / r)
      f (Known 0 _ _ _ _) _ = double 0
      f l (Known 1 _ _ _ _) = l 
      f l r | isOpen l || isOpen r = Open (\h -> f (app_open l h) (app_open r h))
      f l r = Unk $ doubleDivide2 (dynamic l) (dynamic r)
  doubleExp = abs (f z)
    where
      f :: P r h M.Double -> P r h M.Double
      f (Known l _ _ _ _) = double (M.exp l) 
      f (Open x) = Open $ f . x
      f (Unk l) = Unk $ doubleExp1 l
  doubleCmp = abs $ abs $ f (s z) z where
    f :: P r h M.Double -> P r h M.Double -> P r h M.Ordering
    f (Known l _ _ _ _) (Known r _ _ _ _) = ordering (M.compare l r)
    f l r | isOpen l || isOpen r = Open (\h -> f (app_open l h) (app_open r h))
    f l r = Unk (cmp2 (dynamic l) (dynamic r))

type instance K repr h M.Float = M.Float
instance Float r => Float (P r) where
  float x = static (x, float x)
  floatPlus = abs (abs (f (s z) z))
    where
      f :: P r h M.Float -> P r h M.Float -> P r h M.Float
      f (Known l _ _ _ _) (Known r _ _ _ _) = float (l + r)
      f (Known 0 _ _ _ _) r = r
      f l (Known 0 _ _ _ _) = l
      f l r | isOpen l || isOpen r = Open (\h -> f (app_open l h) (app_open r h))
      f l r = Unk (floatPlus2 (dynamic l) (dynamic r))
  floatMult = abs (abs (f (s z) z))
    where
      f :: P r h M.Float -> P r h M.Float -> P r h M.Float
      f (Known l _ _ _ _) (Known r _ _ _ _) = float (l * r)
      f (Known 0 _ _ _ _) _ = float 0
      f _ (Known 0 _ _ _ _) = float 0
      f l (Known 1 _ _ _ _) = l
      f (Known 1 _ _ _ _) r = r
      f l r | isOpen l || isOpen r = Open (\h -> f (app_open l h) (app_open r h))
      f l r = Unk (floatMult2 (dynamic l) (dynamic r))
  floatMinus = abs (abs (f (s z) z))
    where
      f :: P r h M.Float -> P r h M.Float -> P r h M.Float
      f (Known l _ _ _ _) (Known r _ _ _ _) = float (l - r)
      f l (Known 0 _ _ _ _) = l 
      f l r | isOpen l || isOpen r = Open (\h -> f (app_open l h) (app_open r h))
      f l r = Unk (floatMinus2 (dynamic l) (dynamic r))
  floatDivide = abs (abs (f (s z) z))
    where
      f :: P r h M.Float -> P r h M.Float -> P r h M.Float
      f (Known l _ _ _ _) (Known r _ _ _ _) = float (l / r)
      f (Known 0 _ _ _ _) _ = float 0
      f l (Known 1 _ _ _ _) = l 
      f l r | isOpen l || isOpen r = Open (\h -> f (app_open l h) (app_open r h))
      f l r = Unk (floatDivide2 (dynamic l) (dynamic r))
  floatExp = abs (f z)
    where
      f :: P r h M.Float -> P r h M.Float
      f (Known l _ _ _ _) = float (M.exp l) 
      f (Open x) = Open $ f . x
      f (Unk l) = Unk (floatExp1 l)

type instance K repr h (a, b) = (P repr h a, P repr h b)
instance Prod r => Prod (P r) where
  mkProd = abs (abs (f (s z) z))
    where
      f :: P r h a -> P r h b -> P r h (a, b)
      f l r = know (l, r)
                (mkProd2 (dynamic l) (dynamic r))
                (\h -> mkProd2 (app_open l h) (app_open r h))
                (mkProd2 (s l) (s r))
  zro = abs (f z)
    where
      f :: P r h (a, b) -> P r h a
      f (Known (l, _) _ _ _ _) = l
      f (Open x) = Open $ f . x
      f (Unk p) = Unk (zro1 p)
  fst = abs (f z)
    where
      f :: P r h (a, b) -> P r h b
      f (Known (_, r) _ _ _ _) = r
      f (Open x) = Open $ f . x
      f (Unk p) = Unk (fst1 p)

type instance K repr h (M.Either a b) = M.Either (P repr h a) (P repr h b)
instance Sum r => Sum (P r) where
  left = abs (f z)
    where
      f :: P r h a -> P r h (M.Either a b)
      f x = know (Left x)
              (left1 $ dynamic x)
              (\h -> left1 $ app_open x h)
              (left1 $ s x)
  right = abs (f z)
    where
      f :: P r h b -> P r h (M.Either a b)
      f x = know (Right x)
              (right1 $ dynamic x)
              (\h -> right1 $ app_open x h)
              (right1 $ s x)
  sumMatch = abs $ abs $ abs (f (s (s z)) (s z) z)
    where
      f :: P r h (a -> c) -> P r h (b -> c) -> P r h (M.Either a b) -> P r h c
      f l _ (Known (M.Left x) _ _ _ _) = app l x
      f _ r (Known (M.Right x) _ _ _ _) = app r x
      f l r (Open x) = Open $ \h -> f (app_open l h) (app_open r h) (x h)
      f l r (Unk x) = Unk $ sumMatch3 (dynamic l) (dynamic r) x

instance Y r => Y (P r) where

type instance K repr h [a] = Maybe (P repr h a, P repr h [a])
instance List repr => List (P repr) where
  nil = static (Nothing, nil)
  cons = abs $ abs (f (s z) z)
    where
      f :: P repr h a -> P repr h [a] -> P repr h [a]
      f h t = know (Just (h, t))
                (cons2 (dynamic h) (dynamic t))
                (\env -> cons2 (app_open h env) (app_open t env))
                (cons2 (s h) (s t))
  listMatch = abs $ abs $ abs (f (s $ s z) (s z) z)
    where
      f :: P repr h b -> P repr h (a -> [a] -> b) -> P repr h [a] -> P repr h b
  You know nothing ,   .
      f _ r (Known (Just (h, t)) _ _ _ _) = app2 r h t
      f l r (Open x) = Open $ \h -> f (app_open l h) (app_open r h) (x h)
      f l r (Unk x) = Unk $ listMatch3 (dynamic l) (dynamic r) x
  listAppend = abs $ abs (f (s z) z)
    where
      f :: P repr h [a] -> P repr h [a] -> P repr h [a]
      f (Known Nothing _ _ _ _) r = r
      f (Known (Just (h, t)) _ _ _ _) r = cons2 h (listAppend2 t r)
      f l (Known Nothing _ _ _ _) = l
      f l r | isOpen l || isOpen r = Open $ \h -> f (app_open l h) (app_open r h)
      f l r = Unk (listAppend2 (dynamic l) (dynamic r))

type instance K repr h (Maybe a) = Maybe (P repr h a)
instance Option repr => Option (P repr) where
  nothing = static (Nothing, nothing)
  just = abs (f z)
    where
      f :: P repr h a -> P repr h (Maybe a)
      f x = know (Just x)
              (just1 $ dynamic x)
              (\h -> just1 $ app_open x h)
              (just1 $ s x)
  optionMatch = abs $ abs $ abs (f (s (s z)) (s z) z)
    where
      f :: P repr h b -> P repr h (a -> b) -> P repr h (Maybe a) -> P repr h b
      f l _ (Known Nothing _ _ _ _) = l
      f _ r (Known (Just x) _ _ _ _) = app r x
      f l r (Open x) = Open $ \h -> f (app_open l h) (app_open r h) (x h)
      f l r (Unk x) = Unk $ optionMatch3 (dynamic l) (dynamic r) x

type instance K repr h M.Char = M.Char
instance Char repr => Char (P repr) where
  char x = static (x, char x)

type instance K repr h M.Ordering = M.Ordering
instance Ordering repr => Ordering (P repr) where
  ordering x = static (x, ordering x)
  sel = abs $ abs $ abs $ abs $ f (s (s (s z))) (s (s z)) (s z) z where
    f :: P repr h a -> P repr h a -> P repr h a -> P repr h M.Ordering -> P repr h a
    f a _ _ (Known M.LT _ _ _ _) = a
    f _ b _ (Known M.EQ _ _ _ _) = b
    f _ _ c (Known M.GT _ _ _ _) = c
    f a b c (Open x) = Open $ \h -> f (app_open a h) (app_open b h) (app_open c h) (x h)
    f a b c (Unk x) = Unk $ sel4 (dynamic a) (dynamic b) (dynamic c) x

type instance K repr h M.Int = M.Int
instance Int repr => Int (P repr) where
  int x = static (x, int x)
  pred = abs (f z) where
    f :: P repr h M.Int -> P repr h M.Int
    f (Known i _ _ _ _) = int $ i - 1
    f (Open x) = Open $ f . x
    f (Unk x) = Unk $ pred1 x
  intCmp = abs $ abs $ f (s z) z where
    f :: P repr h M.Int -> P repr h M.Int -> P repr h M.Ordering
    f (Known l _ _ _ _) (Known r _ _ _ _) = ordering $ M.compare l r
    f l r | isOpen l || isOpen r = Open $ \h -> f (app_open l h) (app_open r h)
    f l r = Unk $ cmp2 (dynamic l) (dynamic r)

type instance K repr h (M.Dual l r) = (P repr h l, P repr h r)
instance Dual repr => Dual (P repr) where
  dual = abs $ f z
    where
      f :: P repr h (a, b) -> P repr h (M.Dual a b)
      f (Known (l, r) _ _ _ _) =
        know (l, r)
          (mkDual2 (dynamic l) (dynamic r))
          (\h -> mkDual2 (app_open l h) (app_open r h))
          (s $ mkDual2 l r)
      f (Open x) = Open $ f . x
      f (Unk x) = Unk $ dual1 x
  runDual = abs (f z)
    where
      f :: P repr h (M.Dual a b) -> P repr h (a, b)
      f (Known (l, r) _ _ _ _) =
        know (l, r)
          (mkProd2 (dynamic l) (dynamic r))
          (\h -> mkProd2 (app_open l h) (app_open r h))
          (mkProd2 (s l) (s r))
      f (Open x) = Open $ f . x
      f (Unk x) = Unk $ runDual1 x
  dualGetOrdC :: forall x y. Ord (P repr) x :- OrdC (P repr) (M.Dual x y)
  dualGetOrdC = Sub (withDict (getOrdC @(P repr) @x Proxy) Dict)

type instance K r h () = ()
instance Unit r => Unit (P r) where
  unit = static ((), unit)

type instance K r h (M.IO a) = P r h a
instance IO r => IO (P r) where
  putStrLn = Unk putStrLn
  ioJoin = abs $ f z where
    f :: P r h (M.IO (M.IO a)) -> P r h (M.IO a)
    f (Known l _ _ _ _) = l
    f (Open x) = Open $ f . x
    f (Unk x) = Unk $ join1 x
  ioMap = abs $ abs $ f (s z) z where
    f :: P r h (a -> b) -> P r h (M.IO a) -> P r h (M.IO b)
    f l (Known a _ _ _ _) = pure1 $ app l a
    f l (Open x) = Open $ \h -> f (app_open l h) (x h)
    f l (Unk x) = Unk $ map2 (dynamic l) x
  ioPure = abs $ f z where
    f :: P r h a -> P r h (M.IO a)
    f x = know x (pure1 $ dynamic x) (\h -> pure1 $ app_open x h) (pure1 $ s x)
  ioAP = abs $ abs $ f (s z) z where
    f :: P r h (M.IO (a -> b)) -> P r h (M.IO a) -> P r h (M.IO b)
    f (Known l _ _ _ _) (Known r _ _ _ _) = pure1 $ app l r
    f l r | isOpen l || isOpen r = Open $ \h -> f (app_open l h) (app_open r h)
    f l r = Unk $ ap2 (dynamic l) (dynamic r)

pe :: DBI repr => P repr () a -> repr () a
pe = dynamic
