{-# LANGUAGE GADTs              #-}
{-# LANGUAGE RankNTypes         #-}
{-# LANGUAGE Safe #-}

--------------------------------------------------------------------------------
-- |
-- A faster free applicative.
-- Based on <https://www.eyrie.org/~zednenem/2013/05/27/freeapp Dave Menendez's work>.
--------------------------------------------------------------------------------
module Control.Applicative.Free.Fast
  (
  -- * The Sequence of Effects
    ASeq(..)
  , reduceASeq
  , hoistASeq
  , traverseASeq
  , rebaseASeq
  -- * The Faster Free Applicative
  , Ap(..)
  , liftAp
  , retractAp
  , runAp
  , runAp_
  , hoistAp
  ) where

import           Control.Applicative
import           Data.Functor.Apply

-- | The free applicative is composed of a sequence of effects,
-- and a pure function to apply that sequence to.
-- The fast free applicative separates these from each other,
-- so that the sequence may be built up independently,
-- and so that 'fmap' can run in constant time by having immediate access to the pure function.
data ASeq f a where
  ANil :: ASeq f ()
  ACons :: f a -> ASeq f u -> ASeq f (a,u)

-- | Interprets the sequence of effects using the semantics for
--   `pure` and `<*>` given by the Applicative instance for 'f'.
reduceASeq :: Applicative f => ASeq f u -> f u
reduceASeq :: forall (f :: * -> *) u. Applicative f => ASeq f u -> f u
reduceASeq ASeq f u
ANil         = u -> f u
forall a. a -> f a
forall (f :: * -> *) a. Applicative f => a -> f a
pure ()
reduceASeq (ACons f a
x ASeq f u
xs) = (,) (a -> u -> u) -> f a -> f (u -> u)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> f a
x f (u -> u) -> f u -> f u
forall a b. f (a -> b) -> f a -> f b
forall (f :: * -> *) a b. Applicative f => f (a -> b) -> f a -> f b
<*> ASeq f u -> f u
forall (f :: * -> *) u. Applicative f => ASeq f u -> f u
reduceASeq ASeq f u
xs

-- | Given a natural transformation from @f@ to @g@ this gives a natural transformation from @ASeq f@ to @ASeq g@.
hoistASeq :: (forall x. f x -> g x) -> ASeq f a -> ASeq g a
hoistASeq :: forall (f :: * -> *) (g :: * -> *) a.
(forall x. f x -> g x) -> ASeq f a -> ASeq g a
hoistASeq forall x. f x -> g x
_ ASeq f a
ANil = ASeq g a
ASeq g ()
forall (f :: * -> *). ASeq f ()
ANil
hoistASeq forall x. f x -> g x
u (ACons f a
x ASeq f u
xs) = g a -> ASeq g u -> ASeq g (a, u)
forall (f :: * -> *) a u. f a -> ASeq f u -> ASeq f (a, u)
ACons (f a -> g a
forall x. f x -> g x
u f a
x) (f x -> g x
forall x. f x -> g x
u (forall x. f x -> g x) -> ASeq f u -> ASeq g u
forall (f :: * -> *) (g :: * -> *) a.
(forall x. f x -> g x) -> ASeq f a -> ASeq g a
`hoistASeq` ASeq f u
xs)

-- | Traverse a sequence with resepect to its interpretation type 'f'.
traverseASeq :: Applicative h => (forall x. f x -> h (g x)) -> ASeq f a -> h (ASeq g a)
traverseASeq :: forall (h :: * -> *) (f :: * -> *) (g :: * -> *) a.
Applicative h =>
(forall x. f x -> h (g x)) -> ASeq f a -> h (ASeq g a)
traverseASeq forall x. f x -> h (g x)
_ ASeq f a
ANil      = ASeq g a -> h (ASeq g a)
forall a. a -> h a
forall (f :: * -> *) a. Applicative f => a -> f a
pure ASeq g a
ASeq g ()
forall (f :: * -> *). ASeq f ()
ANil
traverseASeq forall x. f x -> h (g x)
f (ACons f a
x ASeq f u
xs) = g a -> ASeq g u -> ASeq g a
g a -> ASeq g u -> ASeq g (a, u)
forall (f :: * -> *) a u. f a -> ASeq f u -> ASeq f (a, u)
ACons (g a -> ASeq g u -> ASeq g a)
-> h (g a) -> h (ASeq g u -> ASeq g a)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> f a -> h (g a)
forall x. f x -> h (g x)
f f a
x h (ASeq g u -> ASeq g a) -> h (ASeq g u) -> h (ASeq g a)
forall a b. h (a -> b) -> h a -> h b
forall (f :: * -> *) a b. Applicative f => f (a -> b) -> f a -> f b
<*> (forall x. f x -> h (g x)) -> ASeq f u -> h (ASeq g u)
forall (h :: * -> *) (f :: * -> *) (g :: * -> *) a.
Applicative h =>
(forall x. f x -> h (g x)) -> ASeq f a -> h (ASeq g a)
traverseASeq f x -> h (g x)
forall x. f x -> h (g x)
f ASeq f u
xs

-- | It may not be obvious, but this essentially acts like ++,
-- traversing the first sequence and creating a new one by appending the second sequence.
-- The difference is that this also has to modify the return functions and that the return type depends on the input types.
--
-- See the source of 'hoistAp' as an example usage.
rebaseASeq :: ASeq f u -> (forall x. (x -> y) -> ASeq f x -> z) ->
  (v -> u -> y) -> ASeq f v -> z
rebaseASeq :: forall (f :: * -> *) u y z v.
ASeq f u
-> (forall x. (x -> y) -> ASeq f x -> z)
-> (v -> u -> y)
-> ASeq f v
-> z
rebaseASeq ASeq f u
ANil         forall x. (x -> y) -> ASeq f x -> z
k v -> u -> y
f = (v -> y) -> ASeq f v -> z
forall x. (x -> y) -> ASeq f x -> z
k (\v
v -> v -> u -> y
f v
v ())
rebaseASeq (ACons f a
x ASeq f u
xs) forall x. (x -> y) -> ASeq f x -> z
k v -> u -> y
f =
  ASeq f u
-> (forall x. (x -> a -> y) -> ASeq f x -> z)
-> (v -> u -> a -> y)
-> ASeq f v
-> z
forall (f :: * -> *) u y z v.
ASeq f u
-> (forall x. (x -> y) -> ASeq f x -> z)
-> (v -> u -> y)
-> ASeq f v
-> z
rebaseASeq ASeq f u
xs (\x -> a -> y
g ASeq f x
s -> ((a, x) -> y) -> ASeq f (a, x) -> z
forall x. (x -> y) -> ASeq f x -> z
k (\(a
a,x
u) -> x -> a -> y
g x
u a
a) (f a -> ASeq f x -> ASeq f (a, x)
forall (f :: * -> *) a u. f a -> ASeq f u -> ASeq f (a, u)
ACons f a
x ASeq f x
s))
    (\v
v u
u a
a -> v -> u -> y
f v
v (a
a,u
u))


-- | The faster free 'Applicative'.
newtype Ap f a = Ap
  { forall (f :: * -> *) a.
Ap f a
-> forall u y z.
   (forall x. (x -> y) -> ASeq f x -> z)
   -> (u -> a -> y) -> ASeq f u -> z
unAp :: forall u y z.
    (forall x. (x -> y) -> ASeq f x -> z) ->
    (u -> a -> y) -> ASeq f u -> z }

-- | Given a natural transformation from @f@ to @g@, this gives a canonical monoidal natural transformation from @'Ap' f@ to @g@.
--
-- prop> runAp t == retractApp . hoistApp t
runAp :: Applicative g => (forall x. f x -> g x) -> Ap f a -> g a
runAp :: forall (g :: * -> *) (f :: * -> *) a.
Applicative g =>
(forall x. f x -> g x) -> Ap f a -> g a
runAp forall x. f x -> g x
u = Ap g a -> g a
forall (f :: * -> *) a. Applicative f => Ap f a -> f a
retractAp (Ap g a -> g a) -> (Ap f a -> Ap g a) -> Ap f a -> g a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (forall x. f x -> g x) -> Ap f a -> Ap g a
forall (f :: * -> *) (g :: * -> *) a.
(forall x. f x -> g x) -> Ap f a -> Ap g a
hoistAp f x -> g x
forall x. f x -> g x
u

-- | Perform a monoidal analysis over free applicative value.
--
-- Example:
--
-- @
-- count :: Ap f a -> Int
-- count = getSum . runAp_ (\\_ -> Sum 1)
-- @
runAp_ :: Monoid m => (forall a. f a -> m) -> Ap f b -> m
runAp_ :: forall m (f :: * -> *) b.
Monoid m =>
(forall a. f a -> m) -> Ap f b -> m
runAp_ forall a. f a -> m
f = Const m b -> m
forall {k} a (b :: k). Const a b -> a
getConst (Const m b -> m) -> (Ap f b -> Const m b) -> Ap f b -> m
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (forall x. f x -> Const m x) -> Ap f b -> Const m b
forall (g :: * -> *) (f :: * -> *) a.
Applicative g =>
(forall x. f x -> g x) -> Ap f a -> g a
runAp (m -> Const m x
forall {k} a (b :: k). a -> Const a b
Const (m -> Const m x) -> (f x -> m) -> f x -> Const m x
forall b c a. (b -> c) -> (a -> b) -> a -> c
. f x -> m
forall a. f a -> m
f)

instance Functor (Ap f) where
  fmap :: forall a b. (a -> b) -> Ap f a -> Ap f b
fmap a -> b
g Ap f a
x = (forall u y z.
 (forall x. (x -> y) -> ASeq f x -> z)
 -> (u -> b -> y) -> ASeq f u -> z)
-> Ap f b
forall (f :: * -> *) a.
(forall u y z.
 (forall x. (x -> y) -> ASeq f x -> z)
 -> (u -> a -> y) -> ASeq f u -> z)
-> Ap f a
Ap (\forall x. (x -> y) -> ASeq f x -> z
k u -> b -> y
f -> Ap f a
-> forall u y z.
   (forall x. (x -> y) -> ASeq f x -> z)
   -> (u -> a -> y) -> ASeq f u -> z
forall (f :: * -> *) a.
Ap f a
-> forall u y z.
   (forall x. (x -> y) -> ASeq f x -> z)
   -> (u -> a -> y) -> ASeq f u -> z
unAp Ap f a
x (x -> y) -> ASeq f x -> z
forall x. (x -> y) -> ASeq f x -> z
k (\u
s -> u -> b -> y
f u
s (b -> y) -> (a -> b) -> a -> y
forall b c a. (b -> c) -> (a -> b) -> a -> c
. a -> b
g))

instance Apply (Ap f) where
  <.> :: forall a b. Ap f (a -> b) -> Ap f a -> Ap f b
(<.>) = Ap f (a -> b) -> Ap f a -> Ap f b
forall a b. Ap f (a -> b) -> Ap f a -> Ap f b
forall (f :: * -> *) a b. Applicative f => f (a -> b) -> f a -> f b
(<*>)

instance Applicative (Ap f) where
  pure :: forall a. a -> Ap f a
pure a
a = (forall u y z.
 (forall x. (x -> y) -> ASeq f x -> z)
 -> (u -> a -> y) -> ASeq f u -> z)
-> Ap f a
forall (f :: * -> *) a.
(forall u y z.
 (forall x. (x -> y) -> ASeq f x -> z)
 -> (u -> a -> y) -> ASeq f u -> z)
-> Ap f a
Ap (\forall x. (x -> y) -> ASeq f x -> z
k u -> a -> y
f -> (u -> y) -> ASeq f u -> z
forall x. (x -> y) -> ASeq f x -> z
k (u -> a -> y
`f` a
a))
  Ap f (a -> b)
x <*> :: forall a b. Ap f (a -> b) -> Ap f a -> Ap f b
<*> Ap f a
y = (forall u y z.
 (forall x. (x -> y) -> ASeq f x -> z)
 -> (u -> b -> y) -> ASeq f u -> z)
-> Ap f b
forall (f :: * -> *) a.
(forall u y z.
 (forall x. (x -> y) -> ASeq f x -> z)
 -> (u -> a -> y) -> ASeq f u -> z)
-> Ap f a
Ap (\forall x. (x -> y) -> ASeq f x -> z
k u -> b -> y
f -> Ap f a
-> forall u y z.
   (forall x. (x -> y) -> ASeq f x -> z)
   -> (u -> a -> y) -> ASeq f u -> z
forall (f :: * -> *) a.
Ap f a
-> forall u y z.
   (forall x. (x -> y) -> ASeq f x -> z)
   -> (u -> a -> y) -> ASeq f u -> z
unAp Ap f a
y (Ap f (a -> b)
-> forall u y z.
   (forall x. (x -> y) -> ASeq f x -> z)
   -> (u -> (a -> b) -> y) -> ASeq f u -> z
forall (f :: * -> *) a.
Ap f a
-> forall u y z.
   (forall x. (x -> y) -> ASeq f x -> z)
   -> (u -> a -> y) -> ASeq f u -> z
unAp Ap f (a -> b)
x (x -> y) -> ASeq f x -> z
forall x. (x -> y) -> ASeq f x -> z
k) (\u
s a
a a -> b
g -> u -> b -> y
f u
s (a -> b
g a
a)))

-- | A version of 'lift' that can be used with just a 'Functor' for @f@.
liftAp :: f a -> Ap f a
liftAp :: forall (f :: * -> *) a. f a -> Ap f a
liftAp f a
a = (forall u y z.
 (forall x. (x -> y) -> ASeq f x -> z)
 -> (u -> a -> y) -> ASeq f u -> z)
-> Ap f a
forall (f :: * -> *) a.
(forall u y z.
 (forall x. (x -> y) -> ASeq f x -> z)
 -> (u -> a -> y) -> ASeq f u -> z)
-> Ap f a
Ap (\forall x. (x -> y) -> ASeq f x -> z
k u -> a -> y
f ASeq f u
s -> ((a, u) -> y) -> ASeq f (a, u) -> z
forall x. (x -> y) -> ASeq f x -> z
k (\(a
a',u
s') -> u -> a -> y
f u
s' a
a') (f a -> ASeq f u -> ASeq f (a, u)
forall (f :: * -> *) a u. f a -> ASeq f u -> ASeq f (a, u)
ACons f a
a ASeq f u
s))
{-# INLINE liftAp #-}

-- | Given a natural transformation from @f@ to @g@ this gives a monoidal natural transformation from @Ap f@ to @Ap g@.
hoistAp :: (forall x. f x -> g x) -> Ap f a -> Ap g a
hoistAp :: forall (f :: * -> *) (g :: * -> *) a.
(forall x. f x -> g x) -> Ap f a -> Ap g a
hoistAp forall x. f x -> g x
g Ap f a
x = (forall u y z.
 (forall x. (x -> y) -> ASeq g x -> z)
 -> (u -> a -> y) -> ASeq g u -> z)
-> Ap g a
forall (f :: * -> *) a.
(forall u y z.
 (forall x. (x -> y) -> ASeq f x -> z)
 -> (u -> a -> y) -> ASeq f u -> z)
-> Ap f a
Ap (\forall x. (x -> y) -> ASeq g x -> z
k u -> a -> y
f ASeq g u
s ->
  Ap f a
-> forall u y z.
   (forall x. (x -> y) -> ASeq f x -> z)
   -> (u -> a -> y) -> ASeq f u -> z
forall (f :: * -> *) a.
Ap f a
-> forall u y z.
   (forall x. (x -> y) -> ASeq f x -> z)
   -> (u -> a -> y) -> ASeq f u -> z
unAp Ap f a
x
    (\x -> a
f' ASeq f x
s' ->
      ASeq g x
-> (forall x. (x -> y) -> ASeq g x -> z)
-> (u -> x -> y)
-> ASeq g u
-> z
forall (f :: * -> *) u y z v.
ASeq f u
-> (forall x. (x -> y) -> ASeq f x -> z)
-> (v -> u -> y)
-> ASeq f v
-> z
rebaseASeq ((forall x. f x -> g x) -> ASeq f x -> ASeq g x
forall (f :: * -> *) (g :: * -> *) a.
(forall x. f x -> g x) -> ASeq f a -> ASeq g a
hoistASeq f x -> g x
forall x. f x -> g x
g ASeq f x
s') (x -> y) -> ASeq g x -> z
forall x. (x -> y) -> ASeq g x -> z
k
        (\u
v x
u -> u -> a -> y
f u
v (x -> a
f' x
u)) ASeq g u
s)
    ((a -> a) -> () -> a -> a
forall a b. a -> b -> a
const a -> a
forall a. a -> a
id)
    ASeq f ()
forall (f :: * -> *). ASeq f ()
ANil)

-- | Interprets the free applicative functor over f using the semantics for
--   `pure` and `<*>` given by the Applicative instance for f.
--
--   prop> retractApp == runAp id
retractAp :: Applicative f => Ap f a -> f a
retractAp :: forall (f :: * -> *) a. Applicative f => Ap f a -> f a
retractAp Ap f a
x = Ap f a
-> forall u y z.
   (forall x. (x -> y) -> ASeq f x -> z)
   -> (u -> a -> y) -> ASeq f u -> z
forall (f :: * -> *) a.
Ap f a
-> forall u y z.
   (forall x. (x -> y) -> ASeq f x -> z)
   -> (u -> a -> y) -> ASeq f u -> z
unAp Ap f a
x (\x -> a
f ASeq f x
s -> x -> a
f (x -> a) -> f x -> f a
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> ASeq f x -> f x
forall (f :: * -> *) u. Applicative f => ASeq f u -> f u
reduceASeq ASeq f x
s) (\() -> a -> a
forall a. a -> a
id) ASeq f ()
forall (f :: * -> *). ASeq f ()
ANil