{-# LANGUAGE CPP               #-}
{-# LANGUAGE DeriveFoldable    #-}
{-# LANGUAGE DeriveFunctor     #-}
{-# LANGUAGE DeriveTraversable #-}

{-# LANGUAGE Safe              #-}

#ifndef MIN_VERSION_transformers
#define MIN_VERSION_transformers(x,y,z) 0
#endif

#ifndef MIN_VERSION_transformers_compat
#define MIN_VERSION_transformers_compat(x,y,z) 0
#endif

#if MIN_VERSION_base(4,9,0)
#define LIFTED_FUNCTOR_CLASSES 1

#elif MIN_VERSION_transformers(0,5,0)
#define LIFTED_FUNCTOR_CLASSES 1

#elif MIN_VERSION_transformers_compat(0,5,0) && !MIN_VERSION_transformers(0,4,0)
#define LIFTED_FUNCTOR_CLASSES 1
#endif

module Data.Function.Step (
    -- * Step Function
    -- $setup
    SF (..),
    Bound (..),
    -- * Construction
    constant,
    step,
    fromList,
    -- * Normalisation
    normalise,
    -- * Operators
    (!),
    values,
    -- * Debug
    showSF,
    putSF,
    ) where

import Control.Applicative  (Applicative (pure, (<*>)), liftA2, (<$>))
import Control.DeepSeq      (NFData (..))
import Control.Monad        (ap)
import Data.Functor.Classes
import Data.List            (intercalate)
import Data.Map             (Map)

import Prelude
       (Eq (..), Functor, IO, Maybe (..), Monad (..), Ord (..), Ordering (..),
       Show (..), String, fst, length, map, otherwise, putStrLn, replicate,
       uncurry, ($), (++), (-), (.))

import Data.Foldable    (Foldable, foldr, maximum)
import Data.Monoid      (Monoid (..))
import Data.Semigroup   (Semigroup (..))
import Data.Traversable (Traversable (traverse))

#ifdef LIFTED_FUNCTOR_CLASSES
import Text.Show (showListWith)
#else
import Prelude (showChar, showParen, showString)
#endif

import qualified Data.Map        as Map
import qualified Test.QuickCheck as QC

-- | Step function. Piecewise constant function, having finitely many pieces.
-- See <https://en.wikipedia.org/wiki/Step_function>.
--
-- @'SF' (fromList [('Open' k1, v1), ('Closed' k2, v2)]) v3 :: 'SF' k v@ describes a piecewise constant function \(f : k \to v\):
--
-- \[
-- f\,x = \begin{cases}
-- v_1, \quad x < k_1 \newline
-- v_2, \quad k_1 \le x \le k_2 \newline
-- v_3, \quad k_2 < x
-- \end{cases}
-- \]
--
-- or as you would write in Haskell
--
-- @
-- f x | x <  k1   = v1
--     | x <= k2   = v2
--     | otherwise = v3
-- @
--
-- /Note:/ [total-map](https://hackage.haskell.org/package/total-map-0.0.6/docs/Data-TotalMap.html) package,
-- which provides /function with finite support/.
--
-- Constructor is exposed as you cannot construct non-valid 'SF'.
--
-- === Merging
--
-- You can use 'Applicative' instance to /merge/ 'SF'.
--
-- >>> putSF $ liftA2 (+) (step 0 0 1) (step 1 0 1)
-- \x -> if
--     | x < 0     -> 0
--     | x < 1     -> 1
--     | otherwise -> 2
--
-- Following property holds, i.e. 'SF' and ordinary function 'Applicative' instances
-- are compatible (and '!' is a homomorphism).
--
-- prop> liftA2 (applyFun2 f) g h ! x == liftA2 (applyFun2 f :: A -> B -> C) (g !) (h !) (x :: Int)
--
-- Recall that for ordinary functions @'liftA2' f g h x = f (g x) (h x)@.
--
-- === Dense?
--
-- This dense variant is useful with [dense ordered](https://en.wikipedia.org/wiki/Dense_order) domains, e.g. 'Rational'.
-- 'Integer' is not dense, so you could use "Data.Function.Step.Discrete" variant instead.
--
-- >>> let s = fromList [(Open 0, -1),(Closed 0, 0)] 1 :: SF Rational Int
-- >>> putSF s
-- \x -> if
--     | x <  0 % 1 -> -1
--     | x <= 0 % 1 -> 0
--     | otherwise  -> 1
--
-- >>> import Data.Ratio ((%))
-- >>> map (s !) [-1, -0.5, 0, 0.5, 1]
-- [-1,-1,0,1,1]
--
data SF k v = SF !(Map (Bound k) v) !v
  deriving (SF k v -> SF k v -> Bool
forall a. (a -> a -> Bool) -> (a -> a -> Bool) -> Eq a
forall k v. (Eq k, Eq v) => SF k v -> SF k v -> Bool
/= :: SF k v -> SF k v -> Bool
$c/= :: forall k v. (Eq k, Eq v) => SF k v -> SF k v -> Bool
== :: SF k v -> SF k v -> Bool
$c== :: forall k v. (Eq k, Eq v) => SF k v -> SF k v -> Bool
Eq, SF k v -> SF k v -> Bool
SF k v -> SF k v -> Ordering
forall a.
Eq a
-> (a -> a -> Ordering)
-> (a -> a -> Bool)
-> (a -> a -> Bool)
-> (a -> a -> Bool)
-> (a -> a -> Bool)
-> (a -> a -> a)
-> (a -> a -> a)
-> Ord a
forall {k} {v}. (Ord k, Ord v) => Eq (SF k v)
forall k v. (Ord k, Ord v) => SF k v -> SF k v -> Bool
forall k v. (Ord k, Ord v) => SF k v -> SF k v -> Ordering
forall k v. (Ord k, Ord v) => SF k v -> SF k v -> SF k v
min :: SF k v -> SF k v -> SF k v
$cmin :: forall k v. (Ord k, Ord v) => SF k v -> SF k v -> SF k v
max :: SF k v -> SF k v -> SF k v
$cmax :: forall k v. (Ord k, Ord v) => SF k v -> SF k v -> SF k v
>= :: SF k v -> SF k v -> Bool
$c>= :: forall k v. (Ord k, Ord v) => SF k v -> SF k v -> Bool
> :: SF k v -> SF k v -> Bool
$c> :: forall k v. (Ord k, Ord v) => SF k v -> SF k v -> Bool
<= :: SF k v -> SF k v -> Bool
$c<= :: forall k v. (Ord k, Ord v) => SF k v -> SF k v -> Bool
< :: SF k v -> SF k v -> Bool
$c< :: forall k v. (Ord k, Ord v) => SF k v -> SF k v -> Bool
compare :: SF k v -> SF k v -> Ordering
$ccompare :: forall k v. (Ord k, Ord v) => SF k v -> SF k v -> Ordering
Ord, forall a b. a -> SF k b -> SF k a
forall a b. (a -> b) -> SF k a -> SF k b
forall k a b. a -> SF k b -> SF k a
forall k a b. (a -> b) -> SF k a -> SF k b
forall (f :: * -> *).
(forall a b. (a -> b) -> f a -> f b)
-> (forall a b. a -> f b -> f a) -> Functor f
<$ :: forall a b. a -> SF k b -> SF k a
$c<$ :: forall k a b. a -> SF k b -> SF k a
fmap :: forall a b. (a -> b) -> SF k a -> SF k b
$cfmap :: forall k a b. (a -> b) -> SF k a -> SF k b
Functor, forall a. SF k a -> Bool
forall k a. Eq a => a -> SF k a -> Bool
forall k a. Num a => SF k a -> a
forall k a. Ord a => SF k a -> a
forall m a. Monoid m => (a -> m) -> SF k a -> m
forall k m. Monoid m => SF k m -> m
forall k a. SF k a -> Bool
forall k a. SF k a -> Int
forall k a. SF k a -> [a]
forall a b. (a -> b -> b) -> b -> SF k a -> b
forall k a. (a -> a -> a) -> SF k a -> a
forall k m a. Monoid m => (a -> m) -> SF k a -> m
forall k b a. (b -> a -> b) -> b -> SF k a -> b
forall k a b. (a -> b -> b) -> b -> SF k a -> b
forall (t :: * -> *).
(forall m. Monoid m => t m -> m)
-> (forall m a. Monoid m => (a -> m) -> t a -> m)
-> (forall m a. Monoid m => (a -> m) -> t a -> m)
-> (forall a b. (a -> b -> b) -> b -> t a -> b)
-> (forall a b. (a -> b -> b) -> b -> t a -> b)
-> (forall b a. (b -> a -> b) -> b -> t a -> b)
-> (forall b a. (b -> a -> b) -> b -> t a -> b)
-> (forall a. (a -> a -> a) -> t a -> a)
-> (forall a. (a -> a -> a) -> t a -> a)
-> (forall a. t a -> [a])
-> (forall a. t a -> Bool)
-> (forall a. t a -> Int)
-> (forall a. Eq a => a -> t a -> Bool)
-> (forall a. Ord a => t a -> a)
-> (forall a. Ord a => t a -> a)
-> (forall a. Num a => t a -> a)
-> (forall a. Num a => t a -> a)
-> Foldable t
product :: forall a. Num a => SF k a -> a
$cproduct :: forall k a. Num a => SF k a -> a
sum :: forall a. Num a => SF k a -> a
$csum :: forall k a. Num a => SF k a -> a
minimum :: forall a. Ord a => SF k a -> a
$cminimum :: forall k a. Ord a => SF k a -> a
maximum :: forall a. Ord a => SF k a -> a
$cmaximum :: forall k a. Ord a => SF k a -> a
elem :: forall a. Eq a => a -> SF k a -> Bool
$celem :: forall k a. Eq a => a -> SF k a -> Bool
length :: forall a. SF k a -> Int
$clength :: forall k a. SF k a -> Int
null :: forall a. SF k a -> Bool
$cnull :: forall k a. SF k a -> Bool
toList :: forall a. SF k a -> [a]
$ctoList :: forall k a. SF k a -> [a]
foldl1 :: forall a. (a -> a -> a) -> SF k a -> a
$cfoldl1 :: forall k a. (a -> a -> a) -> SF k a -> a
foldr1 :: forall a. (a -> a -> a) -> SF k a -> a
$cfoldr1 :: forall k a. (a -> a -> a) -> SF k a -> a
foldl' :: forall b a. (b -> a -> b) -> b -> SF k a -> b
$cfoldl' :: forall k b a. (b -> a -> b) -> b -> SF k a -> b
foldl :: forall b a. (b -> a -> b) -> b -> SF k a -> b
$cfoldl :: forall k b a. (b -> a -> b) -> b -> SF k a -> b
foldr' :: forall a b. (a -> b -> b) -> b -> SF k a -> b
$cfoldr' :: forall k a b. (a -> b -> b) -> b -> SF k a -> b
foldr :: forall a b. (a -> b -> b) -> b -> SF k a -> b
$cfoldr :: forall k a b. (a -> b -> b) -> b -> SF k a -> b
foldMap' :: forall m a. Monoid m => (a -> m) -> SF k a -> m
$cfoldMap' :: forall k m a. Monoid m => (a -> m) -> SF k a -> m
foldMap :: forall m a. Monoid m => (a -> m) -> SF k a -> m
$cfoldMap :: forall k m a. Monoid m => (a -> m) -> SF k a -> m
fold :: forall m. Monoid m => SF k m -> m
$cfold :: forall k m. Monoid m => SF k m -> m
Foldable, forall k. Functor (SF k)
forall k. Foldable (SF k)
forall k (m :: * -> *) a. Monad m => SF k (m a) -> m (SF k a)
forall k (f :: * -> *) a. Applicative f => SF k (f a) -> f (SF k a)
forall k (m :: * -> *) a b.
Monad m =>
(a -> m b) -> SF k a -> m (SF k b)
forall k (f :: * -> *) a b.
Applicative f =>
(a -> f b) -> SF k a -> f (SF k b)
forall (t :: * -> *).
Functor t
-> Foldable t
-> (forall (f :: * -> *) a b.
    Applicative f =>
    (a -> f b) -> t a -> f (t b))
-> (forall (f :: * -> *) a. Applicative f => t (f a) -> f (t a))
-> (forall (m :: * -> *) a b.
    Monad m =>
    (a -> m b) -> t a -> m (t b))
-> (forall (m :: * -> *) a. Monad m => t (m a) -> m (t a))
-> Traversable t
forall (f :: * -> *) a b.
Applicative f =>
(a -> f b) -> SF k a -> f (SF k b)
sequence :: forall (m :: * -> *) a. Monad m => SF k (m a) -> m (SF k a)
$csequence :: forall k (m :: * -> *) a. Monad m => SF k (m a) -> m (SF k a)
mapM :: forall (m :: * -> *) a b.
Monad m =>
(a -> m b) -> SF k a -> m (SF k b)
$cmapM :: forall k (m :: * -> *) a b.
Monad m =>
(a -> m b) -> SF k a -> m (SF k b)
sequenceA :: forall (f :: * -> *) a. Applicative f => SF k (f a) -> f (SF k a)
$csequenceA :: forall k (f :: * -> *) a. Applicative f => SF k (f a) -> f (SF k a)
traverse :: forall (f :: * -> *) a b.
Applicative f =>
(a -> f b) -> SF k a -> f (SF k b)
$ctraverse :: forall k (f :: * -> *) a b.
Applicative f =>
(a -> f b) -> SF k a -> f (SF k b)
Traversable)

-- | Bound operations
data Bound k
    = Open k   -- ^ less-than, @<@
    | Closed k -- ^ less-than-or-equal, @≤@.
  deriving (Bound k -> Bound k -> Bool
forall k. Eq k => Bound k -> Bound k -> Bool
forall a. (a -> a -> Bool) -> (a -> a -> Bool) -> Eq a
/= :: Bound k -> Bound k -> Bool
$c/= :: forall k. Eq k => Bound k -> Bound k -> Bool
== :: Bound k -> Bound k -> Bool
$c== :: forall k. Eq k => Bound k -> Bound k -> Bool
Eq, Int -> Bound k -> ShowS
forall k. Show k => Int -> Bound k -> ShowS
forall k. Show k => [Bound k] -> ShowS
forall k. Show k => Bound k -> String
forall a.
(Int -> a -> ShowS) -> (a -> String) -> ([a] -> ShowS) -> Show a
showList :: [Bound k] -> ShowS
$cshowList :: forall k. Show k => [Bound k] -> ShowS
show :: Bound k -> String
$cshow :: forall k. Show k => Bound k -> String
showsPrec :: Int -> Bound k -> ShowS
$cshowsPrec :: forall k. Show k => Int -> Bound k -> ShowS
Show, forall a b. a -> Bound b -> Bound a
forall a b. (a -> b) -> Bound a -> Bound b
forall (f :: * -> *).
(forall a b. (a -> b) -> f a -> f b)
-> (forall a b. a -> f b -> f a) -> Functor f
<$ :: forall a b. a -> Bound b -> Bound a
$c<$ :: forall a b. a -> Bound b -> Bound a
fmap :: forall a b. (a -> b) -> Bound a -> Bound b
$cfmap :: forall a b. (a -> b) -> Bound a -> Bound b
Functor, forall a. Eq a => a -> Bound a -> Bool
forall a. Num a => Bound a -> a
forall a. Ord a => Bound a -> a
forall m. Monoid m => Bound m -> m
forall a. Bound a -> Bool
forall a. Bound a -> Int
forall a. Bound a -> [a]
forall a. (a -> a -> a) -> Bound a -> a
forall m a. Monoid m => (a -> m) -> Bound a -> m
forall b a. (b -> a -> b) -> b -> Bound a -> b
forall a b. (a -> b -> b) -> b -> Bound a -> b
forall (t :: * -> *).
(forall m. Monoid m => t m -> m)
-> (forall m a. Monoid m => (a -> m) -> t a -> m)
-> (forall m a. Monoid m => (a -> m) -> t a -> m)
-> (forall a b. (a -> b -> b) -> b -> t a -> b)
-> (forall a b. (a -> b -> b) -> b -> t a -> b)
-> (forall b a. (b -> a -> b) -> b -> t a -> b)
-> (forall b a. (b -> a -> b) -> b -> t a -> b)
-> (forall a. (a -> a -> a) -> t a -> a)
-> (forall a. (a -> a -> a) -> t a -> a)
-> (forall a. t a -> [a])
-> (forall a. t a -> Bool)
-> (forall a. t a -> Int)
-> (forall a. Eq a => a -> t a -> Bool)
-> (forall a. Ord a => t a -> a)
-> (forall a. Ord a => t a -> a)
-> (forall a. Num a => t a -> a)
-> (forall a. Num a => t a -> a)
-> Foldable t
product :: forall a. Num a => Bound a -> a
$cproduct :: forall a. Num a => Bound a -> a
sum :: forall a. Num a => Bound a -> a
$csum :: forall a. Num a => Bound a -> a
minimum :: forall a. Ord a => Bound a -> a
$cminimum :: forall a. Ord a => Bound a -> a
maximum :: forall a. Ord a => Bound a -> a
$cmaximum :: forall a. Ord a => Bound a -> a
elem :: forall a. Eq a => a -> Bound a -> Bool
$celem :: forall a. Eq a => a -> Bound a -> Bool
length :: forall a. Bound a -> Int
$clength :: forall a. Bound a -> Int
null :: forall a. Bound a -> Bool
$cnull :: forall a. Bound a -> Bool
toList :: forall a. Bound a -> [a]
$ctoList :: forall a. Bound a -> [a]
foldl1 :: forall a. (a -> a -> a) -> Bound a -> a
$cfoldl1 :: forall a. (a -> a -> a) -> Bound a -> a
foldr1 :: forall a. (a -> a -> a) -> Bound a -> a
$cfoldr1 :: forall a. (a -> a -> a) -> Bound a -> a
foldl' :: forall b a. (b -> a -> b) -> b -> Bound a -> b
$cfoldl' :: forall b a. (b -> a -> b) -> b -> Bound a -> b
foldl :: forall b a. (b -> a -> b) -> b -> Bound a -> b
$cfoldl :: forall b a. (b -> a -> b) -> b -> Bound a -> b
foldr' :: forall a b. (a -> b -> b) -> b -> Bound a -> b
$cfoldr' :: forall a b. (a -> b -> b) -> b -> Bound a -> b
foldr :: forall a b. (a -> b -> b) -> b -> Bound a -> b
$cfoldr :: forall a b. (a -> b -> b) -> b -> Bound a -> b
foldMap' :: forall m a. Monoid m => (a -> m) -> Bound a -> m
$cfoldMap' :: forall m a. Monoid m => (a -> m) -> Bound a -> m
foldMap :: forall m a. Monoid m => (a -> m) -> Bound a -> m
$cfoldMap :: forall m a. Monoid m => (a -> m) -> Bound a -> m
fold :: forall m. Monoid m => Bound m -> m
$cfold :: forall m. Monoid m => Bound m -> m
Foldable, Functor Bound
Foldable Bound
forall (t :: * -> *).
Functor t
-> Foldable t
-> (forall (f :: * -> *) a b.
    Applicative f =>
    (a -> f b) -> t a -> f (t b))
-> (forall (f :: * -> *) a. Applicative f => t (f a) -> f (t a))
-> (forall (m :: * -> *) a b.
    Monad m =>
    (a -> m b) -> t a -> m (t b))
-> (forall (m :: * -> *) a. Monad m => t (m a) -> m (t a))
-> Traversable t
forall (m :: * -> *) a. Monad m => Bound (m a) -> m (Bound a)
forall (f :: * -> *) a. Applicative f => Bound (f a) -> f (Bound a)
forall (m :: * -> *) a b.
Monad m =>
(a -> m b) -> Bound a -> m (Bound b)
forall (f :: * -> *) a b.
Applicative f =>
(a -> f b) -> Bound a -> f (Bound b)
sequence :: forall (m :: * -> *) a. Monad m => Bound (m a) -> m (Bound a)
$csequence :: forall (m :: * -> *) a. Monad m => Bound (m a) -> m (Bound a)
mapM :: forall (m :: * -> *) a b.
Monad m =>
(a -> m b) -> Bound a -> m (Bound b)
$cmapM :: forall (m :: * -> *) a b.
Monad m =>
(a -> m b) -> Bound a -> m (Bound b)
sequenceA :: forall (f :: * -> *) a. Applicative f => Bound (f a) -> f (Bound a)
$csequenceA :: forall (f :: * -> *) a. Applicative f => Bound (f a) -> f (Bound a)
traverse :: forall (f :: * -> *) a b.
Applicative f =>
(a -> f b) -> Bound a -> f (Bound b)
$ctraverse :: forall (f :: * -> *) a b.
Applicative f =>
(a -> f b) -> Bound a -> f (Bound b)
Traversable)

-- | Order is like @'Open' k = (k, False)@, @'Closed' k = (k, True)@.
--
instance Ord k => Ord (Bound k) where
    compare :: Bound k -> Bound k -> Ordering
compare (Open k
k)   (Open k
k')   = forall a. Ord a => a -> a -> Ordering
compare k
k k
k'
    compare (Closed k
k) (Closed k
k') = forall a. Ord a => a -> a -> Ordering
compare k
k k
k'
    compare (Open k
k)   (Closed k
k') = case forall a. Ord a => a -> a -> Ordering
compare k
k k
k' of
        Ordering
LT -> Ordering
LT
        Ordering
EQ -> Ordering
LT
        Ordering
GT -> Ordering
GT
    compare (Closed k
k) (Open k
k')   = case forall a. Ord a => a -> a -> Ordering
compare k
k k
k' of
        Ordering
LT -> Ordering
LT
        Ordering
EQ -> Ordering
GT
        Ordering
GT -> Ordering
GT

-------------------------------------------------------------------------------
-- Instances
-------------------------------------------------------------------------------

-- | 'pure' is a constant function.
instance Ord k => Applicative (SF k) where
    pure :: forall a. a -> SF k a
pure  = forall a k. a -> SF k a
constant
    <*> :: forall a b. SF k (a -> b) -> SF k a -> SF k b
(<*>) = forall (m :: * -> *) a b. Monad m => m (a -> b) -> m a -> m b
ap

instance Ord k => Monad (SF k) where
    return :: forall a. a -> SF k a
return = forall (f :: * -> *) a. Applicative f => a -> f a
pure

    SF Map (Bound k) a
m a
def0 >>= :: forall a b. SF k a -> (a -> SF k b) -> SF k b
>>= a -> SF k b
f = forall k v. Map (Bound k) v -> v -> SF k v
SF
        (forall k a. [(k, a)] -> Map k a
Map.fromDistinctAscList forall a b. (a -> b) -> a -> b
$ forall k b. Ord k => [(k, b)] -> [(k, b)]
mkDistinctAscList forall a b. (a -> b) -> a -> b
$ [(Bound k, b)]
pieces forall a. [a] -> [a] -> [a]
++ [(Bound k, b)]
pieces1)
        b
def1
      where
        pieces :: [(Bound k, b)]
pieces =
            [ (forall a. Ord a => a -> a -> a
min Bound k
k Bound k
k', b
v')
            | (Bound k
k, a
v) <- forall k a. Map k a -> [(k, a)]
Map.toList Map (Bound k) a
m
            , let SF Map (Bound k) b
m' b
def = a -> SF k b
f a
v
            , (Bound k
k', b
v') <- forall k a. Map k a -> [(k, a)]
Map.toList Map (Bound k) b
m' forall a. [a] -> [a] -> [a]
++ [(Bound k
k, b
def)]
            ]
        ([(Bound k, b)]
pieces1, b
def1) = let SF Map (Bound k) b
m' b
def = a -> SF k b
f a
def0 in (forall k a. Map k a -> [(k, a)]
Map.toList Map (Bound k) b
m', b
def)

-- | Piecewise '<>'.
--
-- >>> putSF $ step 0 "a" "b" <> step 1 "c" "d"
-- \x -> if
--     | x < 0     -> "ac"
--     | x < 1     -> "bc"
--     | otherwise -> "bd"
--
instance (Ord k, Semigroup v) => Semigroup (SF k v) where
    <> :: SF k v -> SF k v -> SF k v
(<>) = forall (f :: * -> *) a b c.
Applicative f =>
(a -> b -> c) -> f a -> f b -> f c
liftA2 forall a. Semigroup a => a -> a -> a
(<>)

instance (Ord k, Monoid v) => Monoid (SF k v) where
    mempty :: SF k v
mempty = forall (f :: * -> *) a. Applicative f => a -> f a
pure forall a. Monoid a => a
mempty
    mappend :: SF k v -> SF k v -> SF k v
mappend = forall (f :: * -> *) a b c.
Applicative f =>
(a -> b -> c) -> f a -> f b -> f c
liftA2 forall a. Monoid a => a -> a -> a
mappend

instance (Ord k, QC.Arbitrary k, QC.Arbitrary v) => QC.Arbitrary (SF k v) where
    arbitrary :: Gen (SF k v)
arbitrary = forall k v. Ord k => [(Bound k, v)] -> v -> SF k v
fromList forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> forall a. Arbitrary a => Gen a
QC.arbitrary forall (f :: * -> *) a b. Applicative f => f (a -> b) -> f a -> f b
<*> forall a. Arbitrary a => Gen a
QC.arbitrary
    shrink :: SF k v -> [SF k v]
shrink (SF Map (Bound k) v
m v
v) = forall a b c. (a -> b -> c) -> (a, b) -> c
uncurry forall k v. Ord k => [(Bound k, v)] -> v -> SF k v
fromList forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> forall a. Arbitrary a => a -> [a]
QC.shrink (forall k a. Map k a -> [(k, a)]
Map.toList Map (Bound k) v
m, v
v)

instance QC.Arbitrary k => QC.Arbitrary (Bound k) where
    arbitrary :: Gen (Bound k)
arbitrary = forall a. [Gen a] -> Gen a
QC.oneof [forall k. k -> Bound k
Open forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> forall a. Arbitrary a => Gen a
QC.arbitrary, forall k. k -> Bound k
Closed forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> forall a. Arbitrary a => Gen a
QC.arbitrary]

instance NFData k => NFData (Bound k) where
    rnf :: Bound k -> ()
rnf (Open k
k) = forall a. NFData a => a -> ()
rnf k
k
    rnf (Closed k
k) = forall a. NFData a => a -> ()
rnf k
k

instance (NFData k, NFData v) => NFData (SF k v) where
    rnf :: SF k v -> ()
rnf (SF Map (Bound k) v
m v
v) = forall a. NFData a => a -> ()
rnf (Map (Bound k) v
m, v
v)

-------------------------------------------------------------------------------
-- Show
-------------------------------------------------------------------------------

#if LIFTED_FUNCTOR_CLASSES

instance Show2 SF where
    liftShowsPrec2 :: forall a b.
(Int -> a -> ShowS)
-> ([a] -> ShowS)
-> (Int -> b -> ShowS)
-> ([b] -> ShowS)
-> Int
-> SF a b
-> ShowS
liftShowsPrec2 Int -> a -> ShowS
spk [a] -> ShowS
slk Int -> b -> ShowS
spv [b] -> ShowS
slv Int
d (SF Map (Bound a) b
m b
v) = forall a b.
(Int -> a -> ShowS)
-> (Int -> b -> ShowS) -> String -> Int -> a -> b -> ShowS
showsBinaryWith
        (\Int
_ -> forall a. (a -> ShowS) -> [a] -> ShowS
showListWith forall a b. (a -> b) -> a -> b
$ forall (f :: * -> * -> *) a b.
Show2 f =>
(Int -> a -> ShowS)
-> ([a] -> ShowS)
-> (Int -> b -> ShowS)
-> ([b] -> ShowS)
-> Int
-> f a b
-> ShowS
liftShowsPrec2 (forall (f :: * -> *) a.
Show1 f =>
(Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> f a -> ShowS
liftShowsPrec Int -> a -> ShowS
spk [a] -> ShowS
slk) (forall (f :: * -> *) a.
Show1 f =>
(Int -> a -> ShowS) -> ([a] -> ShowS) -> [f a] -> ShowS
liftShowList Int -> a -> ShowS
spk [a] -> ShowS
slk) Int -> b -> ShowS
spv [b] -> ShowS
slv Int
0)
        Int -> b -> ShowS
spv
        String
"fromList" Int
d (forall k a. Map k a -> [(k, a)]
Map.toList Map (Bound a) b
m) b
v

instance Show k => Show1 (SF k) where
    liftShowsPrec :: forall a.
(Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> SF k a -> ShowS
liftShowsPrec = forall (f :: * -> * -> *) a b.
Show2 f =>
(Int -> a -> ShowS)
-> ([a] -> ShowS)
-> (Int -> b -> ShowS)
-> ([b] -> ShowS)
-> Int
-> f a b
-> ShowS
liftShowsPrec2 forall a. Show a => Int -> a -> ShowS
showsPrec forall a. Show a => [a] -> ShowS
showList

instance (Show k, Show v) => Show (SF k v) where
    showsPrec :: Int -> SF k v -> ShowS
showsPrec = forall (f :: * -> * -> *) a b.
(Show2 f, Show a, Show b) =>
Int -> f a b -> ShowS
showsPrec2

instance Show1 Bound where
    liftShowsPrec :: forall a.
(Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> Bound a -> ShowS
liftShowsPrec Int -> a -> ShowS
sp [a] -> ShowS
_ Int
d (Open a
k)   = forall a. (Int -> a -> ShowS) -> String -> Int -> a -> ShowS
showsUnaryWith Int -> a -> ShowS
sp String
"Open"   Int
d a
k
    liftShowsPrec Int -> a -> ShowS
sp [a] -> ShowS
_ Int
d (Closed a
k) = forall a. (Int -> a -> ShowS) -> String -> Int -> a -> ShowS
showsUnaryWith Int -> a -> ShowS
sp String
"Closed" Int
d a
k

#else

instance (Show k, Show v) => Show (SF k v) where
    showsPrec d (SF m v) = showParen (d > 10)
        $ showString "fromList"
        . showsPrec 11 (Map.toList m)
        . showChar ' '
        . showsPrec 11 v

instance Show k => Show1 (SF k) where showsPrec1 = showsPrec
instance Show1 Bound where showsPrec1 = showsPrec

#endif
-------------------------------------------------------------------------------
-- Helpers
-------------------------------------------------------------------------------

mkDistinctAscList :: Ord k => [(k, b)] -> [(k, b)]
mkDistinctAscList :: forall k b. Ord k => [(k, b)] -> [(k, b)]
mkDistinctAscList []            = []
mkDistinctAscList ((k
k, b
v) : [(k, b)]
kv) = (k
k, b
v) forall a. a -> [a] -> [a]
: forall k b. Ord k => k -> [(k, b)] -> [(k, b)]
mkDistinctAscList' k
k [(k, b)]
kv

mkDistinctAscList' :: Ord k => k -> [(k, b)] -> [(k, b)]
mkDistinctAscList' :: forall k b. Ord k => k -> [(k, b)] -> [(k, b)]
mkDistinctAscList' k
_ [] = []
mkDistinctAscList' k
k (p :: (k, b)
p@(k
k', b
_) : [(k, b)]
kv)
    | k
k forall a. Ord a => a -> a -> Bool
< k
k'    = (k, b)
p forall a. a -> [a] -> [a]
: forall k b. Ord k => k -> [(k, b)] -> [(k, b)]
mkDistinctAscList' k
k' [(k, b)]
kv
    | Bool
otherwise =     forall k b. Ord k => k -> [(k, b)] -> [(k, b)]
mkDistinctAscList' k
k  [(k, b)]
kv

-------------------------------------------------------------------------------
-- Operators
-------------------------------------------------------------------------------

infixl 9 !

-- | Apply 'SF'.
--
-- >>> heaviside ! 2
-- 1
(!) :: Ord k => SF k v -> k -> v
SF Map (Bound k) v
m v
def ! :: forall k v. Ord k => SF k v -> k -> v
! k
x = case forall k v. Ord k => k -> Map k v -> Maybe (k, v)
Map.lookupGE (forall k. k -> Bound k
Closed k
x) Map (Bound k) v
m of
    Maybe (Bound k, v)
Nothing     -> v
def
    Just (Bound k
_, v
v) -> v
v

-------------------------------------------------------------------------------
-- Construction
-------------------------------------------------------------------------------

-- | Constant function
--
-- >>> putSF $ constant 1
-- \_ -> 1
--
constant :: a -> SF k a
constant :: forall a k. a -> SF k a
constant = forall k v. Map (Bound k) v -> v -> SF k v
SF forall k a. Map k a
Map.empty

-- | Step function.
--
-- @'step' k v1 v2 = \\ x -> if x < k then v1 else v2@.
--
-- >>> putSF $ step 1 2 3
-- \x -> if
--     | x < 1     -> 2
--     | otherwise -> 3
--
step :: k -> v -> v -> SF k v
step :: forall k v. k -> v -> v -> SF k v
step k
k = forall k v. Map (Bound k) v -> v -> SF k v
SF forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall k a. k -> a -> Map k a
Map.singleton (forall k. k -> Bound k
Open k
k)

-- | Create function from list of cases and default value.
--
-- >>> let f = fromList [(Open 1,2),(Closed 3,4),(Open 4,5)] 6
-- >>> putSF f
-- \x -> if
--     | x <  1    -> 2
--     | x <= 3    -> 4
--     | x <  4    -> 5
--     | otherwise -> 6
--
-- >>> map (f !) [0..10]
-- [2,4,4,4,6,6,6,6,6,6,6]
--
fromList :: Ord k => [(Bound k, v)] -> v -> SF k v
fromList :: forall k v. Ord k => [(Bound k, v)] -> v -> SF k v
fromList = forall k v. Map (Bound k) v -> v -> SF k v
SF forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall k a. Ord k => [(k, a)] -> Map k a
Map.fromList

-------------------------------------------------------------------------------
-- Conversions to/from list
-------------------------------------------------------------------------------

-- | Possible values of 'SF'
--
-- >>> values heaviside
-- [-1,1]
--
values :: SF k v -> [v]
values :: forall k a. SF k a -> [a]
values (SF Map (Bound k) v
m v
v) = forall k a. Map k a -> [a]
Map.elems Map (Bound k) v
m forall a. [a] -> [a] -> [a]
++ [v
v]

-------------------------------------------------------------------------------
-- Normalise
-------------------------------------------------------------------------------

-- | Merge adjustent pieces with same values.
--
-- /Note:/ 'SF' isn't normalised on construction.
-- Values don't necessarily are 'Eq'.
--
-- >>> putSF $ normalise heaviside
-- \x -> if
--     | x < 0     -> -1
--     | otherwise -> 1
--
-- >>> putSF $ normalise $ step 0 1 1
-- \_ -> 1
--
-- prop> normalise (liftA2 (+) p (fmap negate p)) == (pure 0 :: SF Int Int)
--
normalise :: Eq v => SF k v -> SF k v
normalise :: forall v k. Eq v => SF k v -> SF k v
normalise (SF Map (Bound k) v
m v
v) = forall a b c. (a -> b -> c) -> (a, b) -> c
uncurry forall {k} {p}. [(Bound k, v)] -> p -> SF k v
mk forall a b. (a -> b) -> a -> b
$ forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr forall {a} {a}. Eq a => (a, a) -> ([(a, a)], a) -> ([(a, a)], a)
go ([], v
v) (forall k a. Map k a -> [(k, a)]
Map.toList Map (Bound k) v
m) where
    mk :: [(Bound k, v)] -> p -> SF k v
mk [(Bound k, v)]
m' p
_ = forall k v. Map (Bound k) v -> v -> SF k v
SF (forall k a. [(k, a)] -> Map k a
Map.fromDistinctAscList [(Bound k, v)]
m') v
v

    go :: (a, a) -> ([(a, a)], a) -> ([(a, a)], a)
go p :: (a, a)
p@(a
_, a
v') p' :: ([(a, a)], a)
p'@([(a, a)]
m', a
x)
        | a
v' forall a. Eq a => a -> a -> Bool
== a
x   = ([(a, a)], a)
p'
        | Bool
otherwise = ((a, a)
p forall a. a -> [a] -> [a]
: [(a, a)]
m', a
v')

-------------------------------------------------------------------------------
-- Pretty-printing
-------------------------------------------------------------------------------

-- | Show 'SF' as Haskell code
showSF :: (Show a, Show b) => SF a b -> String
showSF :: forall k v. (Show k, Show v) => SF k v -> String
showSF (SF Map (Bound a) b
m b
v) | forall k a. Map k a -> Bool
Map.null Map (Bound a) b
m = String
"\\_ -> " forall a. [a] -> [a] -> [a]
++ forall a. Show a => a -> String
show b
v
showSF (SF Map (Bound a) b
m b
v) = forall a. [a] -> [[a]] -> [a]
intercalate String
"\n" forall a b. (a -> b) -> a -> b
$
    String
"\\x -> if" forall a. a -> [a] -> [a]
: [ String
"    | " forall a. [a] -> [a] -> [a]
++ ShowS
leftPad String
k forall a. [a] -> [a] -> [a]
++ String
" -> " forall a. [a] -> [a] -> [a]
++ String
x | (String
k, String
x) <- [(String, String)]
cases ]
  where
    cases :: [(String, String)]
cases     = [(String, String)]
cases' forall a. [a] -> [a] -> [a]
++ [ (String
"otherwise", forall a. Show a => a -> String
show b
v) ]

    m' :: [(Bound a, b)]
m' = forall k a. Map k a -> [(k, a)]
Map.toList Map (Bound a) b
m

    cases' :: [(String, String)]
cases' = case forall (t :: * -> *) (f :: * -> *) a b.
(Traversable t, Applicative f) =>
(a -> f b) -> t a -> f (t b)
traverse forall {a} {b}. (Bound a, b) -> Maybe (a, b)
fromOpen [(Bound a, b)]
m' of
        Maybe [(a, b)]
Nothing  -> [ (String
"x " forall a. [a] -> [a] -> [a]
++ forall k. Show k => Bound k -> String
showBound Bound a
k, forall a. Show a => a -> String
show b
x) | (Bound a
k, b
x) <- [(Bound a, b)]
m' ]
        Just [(a, b)]
m'' -> [ (String
"x < " forall a. [a] -> [a] -> [a]
++ forall a. Show a => a -> String
show a
k,    forall a. Show a => a -> String
show b
x) | (a
k, b
x) <- [(a, b)]
m'' ]

    fromOpen :: (Bound a, b) -> Maybe (a, b)
fromOpen (Open a
k, b
x) = forall a. a -> Maybe a
Just (a
k, b
x)
    fromOpen (Bound a, b)
_           = forall a. Maybe a
Nothing

    len :: Int
len       = forall (t :: * -> *) a. (Foldable t, Ord a) => t a -> a
maximum (forall a b. (a -> b) -> [a] -> [b]
map (forall (t :: * -> *) a. Foldable t => t a -> Int
length forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a b. (a, b) -> a
fst) [(String, String)]
cases)
    leftPad :: ShowS
leftPad String
s = String
s forall a. [a] -> [a] -> [a]
++ forall a. Int -> a -> [a]
replicate (Int
len forall a. Num a => a -> a -> a
- forall (t :: * -> *) a. Foldable t => t a -> Int
length String
s) Char
' '

showBound :: Show k => Bound k -> String
showBound :: forall k. Show k => Bound k -> String
showBound (Open k
k)   = String
"<  " forall a. [a] -> [a] -> [a]
++ forall a. Show a => Int -> a -> ShowS
showsPrec Int
5 k
k String
""
showBound (Closed k
k) = String
"<= " forall a. [a] -> [a] -> [a]
++ forall a. Show a => Int -> a -> ShowS
showsPrec Int
5 k
k String
""

-- | @'putStrLn' . 'showSF'@
putSF :: (Show a, Show b) => SF a b -> IO ()
putSF :: forall a b. (Show a, Show b) => SF a b -> IO ()
putSF = String -> IO ()
putStrLn forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall k v. (Show k, Show v) => SF k v -> String
showSF

-- $setup
--
-- >>> import Test.QuickCheck (applyFun2)
-- >>> import Test.QuickCheck.Poly (A, B, C)
-- >>> import Control.Applicative (liftA2, pure)
-- >>> import Data.Semigroup (Semigroup (..))
--
-- == Examples
--
-- >>> let heaviside = step 0 (-1) 1 :: SF Int Int
-- >>> putSF heaviside
-- \x -> if
--     | x < 0     -> -1
--     | otherwise -> 1
--
-- >>> map (heaviside !) [-3, 0, 4]
-- [-1,1,1]