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

{-# LANGUAGE Safe              #-}

module Data.Function.Step.Discrete.Open (
    -- * Step Function
    -- $setup
    SF (..),
    -- * Construction
    constant,
    step,
    fromList,
    -- * Normalisation
    normalise,
    -- * Operators
    (!),
    values,
    -- * Conversions
    toDense,
    fromDense,
    -- * 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 Data.Maybe           (mapMaybe)

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

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

import Text.Show (showListWith)

import qualified Data.Function.Step as SF
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>.
--
-- /Note:/ this variant has discrete domain.
-- It's enough to have only @<@$, without @≤@, as there is a /next/ element
-- without any others in between.
--
-- @'SF' (fromList [(k1, v1), (k2, v2)]) v3 :: 'SF' k v@ describes a piecewise constant function \(f : k \to v\):
--
-- \[
-- f\,x = \begin{cases}
-- v_1, \quad x < k_1 \newline
-- v_2, \quad k_1 \le x < k_2 \newline
-- v_3, \quad k_2 \le x
-- \end{cases}
-- \]
--
-- or as you would write in Haskell
--
-- @
-- f x | x < k1    = v1
--     | x < k2    = v2
--     | otherwise = v3
-- @
--
-- Constructor is exposed as you cannot construct non-valid 'SF'.
--
data SF k v = SF !(Map k v) !v
  deriving (SF k v -> SF k v -> Bool
(SF k v -> SF k v -> Bool)
-> (SF k v -> SF k v -> Bool) -> Eq (SF k v)
forall a. (a -> a -> Bool) -> (a -> a -> Bool) -> Eq a
forall k v. (Eq k, Eq v) => 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
/= :: SF k v -> SF k v -> Bool
Eq, Eq (SF k v)
Eq (SF k v) =>
(SF k v -> SF k v -> Ordering)
-> (SF k v -> SF k v -> Bool)
-> (SF k v -> SF k v -> Bool)
-> (SF k v -> SF k v -> Bool)
-> (SF k v -> SF k v -> Bool)
-> (SF k v -> SF k v -> SF k v)
-> (SF k v -> SF k v -> SF k v)
-> Ord (SF k v)
SF k v -> SF k v -> Bool
SF k v -> SF k v -> Ordering
SF k v -> SF k v -> SF k v
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
$ccompare :: forall k v. (Ord k, Ord v) => SF k v -> SF k v -> Ordering
compare :: SF k v -> SF k v -> Ordering
$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
>= :: SF k v -> SF k v -> Bool
$cmax :: 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
$cmin :: 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
Ord, (forall a b. (a -> b) -> SF k a -> SF k b)
-> (forall a b. a -> SF k b -> SF k a) -> Functor (SF k)
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
$cfmap :: forall k a b. (a -> b) -> SF k a -> SF k b
fmap :: forall a b. (a -> b) -> SF k a -> SF k b
$c<$ :: forall k a b. a -> SF k b -> SF k a
<$ :: forall a b. a -> SF k b -> SF k a
Functor, (forall m. Monoid m => SF k m -> m)
-> (forall m a. Monoid m => (a -> m) -> SF k a -> m)
-> (forall m a. Monoid m => (a -> m) -> SF k a -> m)
-> (forall a b. (a -> b -> b) -> b -> SF k a -> b)
-> (forall a b. (a -> b -> b) -> b -> SF k a -> b)
-> (forall b a. (b -> a -> b) -> b -> SF k a -> b)
-> (forall b a. (b -> a -> b) -> b -> SF k a -> b)
-> (forall a. (a -> a -> a) -> SF k a -> a)
-> (forall a. (a -> a -> a) -> SF k a -> a)
-> (forall a. SF k a -> [a])
-> (forall a. SF k a -> Bool)
-> (forall a. SF k a -> Int)
-> (forall a. Eq a => a -> SF k a -> Bool)
-> (forall a. Ord a => SF k a -> a)
-> (forall a. Ord a => SF k a -> a)
-> (forall a. Num a => SF k a -> a)
-> (forall a. Num a => SF k a -> a)
-> Foldable (SF k)
forall a. Eq a => a -> SF k a -> Bool
forall a. Num a => SF k a -> a
forall a. Ord a => SF k a -> a
forall m. Monoid m => SF k m -> m
forall a. SF k a -> Bool
forall a. SF k a -> Int
forall a. SF k a -> [a]
forall a. (a -> a -> a) -> SF k a -> a
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 b a. (b -> a -> b) -> b -> SF k a -> b
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
$cfold :: forall k m. Monoid m => SF k m -> m
fold :: forall m. Monoid m => SF k m -> 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
foldMap' :: forall m a. Monoid m => (a -> m) -> SF k a -> m
$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
foldr' :: forall a b. (a -> b -> 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
foldl' :: forall b a. (b -> a -> b) -> b -> SF k a -> b
$cfoldr1 :: forall k a. (a -> a -> a) -> SF k a -> a
foldr1 :: forall a. (a -> a -> a) -> SF k a -> a
$cfoldl1 :: forall k a. (a -> a -> a) -> SF k a -> a
foldl1 :: forall a. (a -> a -> a) -> SF k a -> a
$ctoList :: forall k a. SF k a -> [a]
toList :: forall a. SF k a -> [a]
$cnull :: forall k a. SF k a -> Bool
null :: forall a. SF k a -> Bool
$clength :: forall k a. SF k a -> Int
length :: forall a. SF k a -> Int
$celem :: forall k a. Eq a => a -> SF k a -> Bool
elem :: forall a. Eq a => a -> SF k a -> Bool
$cmaximum :: forall k a. Ord a => SF k a -> a
maximum :: forall a. Ord a => SF k a -> a
$cminimum :: forall k a. Ord a => SF k a -> a
minimum :: forall a. Ord a => SF k a -> a
$csum :: forall k a. Num a => SF k a -> a
sum :: forall a. Num a => SF k a -> a
$cproduct :: forall k a. Num a => SF k a -> a
product :: forall a. Num a => SF k a -> a
Foldable, Functor (SF k)
Foldable (SF k)
(Functor (SF k), Foldable (SF k)) =>
(forall (f :: * -> *) a b.
 Applicative f =>
 (a -> f b) -> SF k a -> f (SF k b))
-> (forall (f :: * -> *) a.
    Applicative f =>
    SF k (f a) -> f (SF k a))
-> (forall (m :: * -> *) a b.
    Monad m =>
    (a -> m b) -> SF k a -> m (SF k b))
-> (forall (m :: * -> *) a. Monad m => SF k (m a) -> m (SF k a))
-> Traversable (SF k)
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 (m :: * -> *) a. Monad m => SF k (m a) -> m (SF k a)
forall (f :: * -> *) a. Applicative f => SF k (f a) -> f (SF k a)
forall (m :: * -> *) a b.
Monad m =>
(a -> m b) -> SF k a -> m (SF k b)
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)
traverse :: forall (f :: * -> *) a b.
Applicative f =>
(a -> f b) -> SF k a -> f (SF k b)
$csequenceA :: forall k (f :: * -> *) a. Applicative f => SF k (f a) -> f (SF k a)
sequenceA :: forall (f :: * -> *) a. Applicative f => SF k (f a) -> f (SF k a)
$cmapM :: forall k (m :: * -> *) a b.
Monad m =>
(a -> m b) -> SF k a -> m (SF k b)
mapM :: forall (m :: * -> *) a b.
Monad m =>
(a -> m b) -> SF k a -> m (SF k b)
$csequence :: forall k (m :: * -> *) a. Monad m => SF k (m a) -> m (SF k a)
sequence :: forall (m :: * -> *) a. Monad m => SF k (m a) -> m (SF k a)
Traversable)

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

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

    SF Map k a
m a
def0 >>= :: forall a b. SF k a -> (a -> SF k b) -> SF k b
>>= a -> SF k b
f = Map k b -> b -> SF k b
forall k v. Map k v -> v -> SF k v
SF
        ([(k, b)] -> Map k b
forall k a. [(k, a)] -> Map k a
Map.fromDistinctAscList ([(k, b)] -> Map k b) -> [(k, b)] -> Map k b
forall a b. (a -> b) -> a -> b
$ [(k, b)] -> [(k, b)]
forall k b. Ord k => [(k, b)] -> [(k, b)]
mkDistinctAscList ([(k, b)] -> [(k, b)]) -> [(k, b)] -> [(k, b)]
forall a b. (a -> b) -> a -> b
$ [(k, b)]
pieces [(k, b)] -> [(k, b)] -> [(k, b)]
forall a. [a] -> [a] -> [a]
++ [(k, b)]
pieces1)
        b
def1
      where
        pieces :: [(k, b)]
pieces =
            [ (k -> k -> k
forall a. Ord a => a -> a -> a
min k
k k
k', b
v')
            | (k
k, a
v) <- Map k a -> [(k, a)]
forall k a. Map k a -> [(k, a)]
Map.toList Map k a
m
            , let SF Map k b
m' b
def = a -> SF k b
f a
v
            , (k
k', b
v') <- Map k b -> [(k, b)]
forall k a. Map k a -> [(k, a)]
Map.toList Map k b
m' [(k, b)] -> [(k, b)] -> [(k, b)]
forall a. [a] -> [a] -> [a]
++ [(k
k, b
def)]
            ]
        ([(k, b)]
pieces1, b
def1) = let SF Map k b
m' b
def = a -> SF k b
f a
def0 in (Map k b -> [(k, b)]
forall k a. Map k a -> [(k, a)]
Map.toList Map 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
(<>) = (v -> v -> v) -> SF k v -> SF k v -> SF k v
forall a b c. (a -> b -> c) -> SF k a -> SF k b -> SF k c
forall (f :: * -> *) a b c.
Applicative f =>
(a -> b -> c) -> f a -> f b -> f c
liftA2 v -> v -> v
forall a. Semigroup a => a -> a -> a
(<>)

instance (Ord k, Monoid v) => Monoid (SF k v) where
    mempty :: SF k v
mempty = v -> SF k v
forall a. a -> SF k a
forall (f :: * -> *) a. Applicative f => a -> f a
pure v
forall a. Monoid a => a
mempty
    mappend :: SF k v -> SF k v -> SF k v
mappend = (v -> v -> v) -> SF k v -> SF k v -> SF k v
forall a b c. (a -> b -> c) -> SF k a -> SF k b -> SF k c
forall (f :: * -> *) a b c.
Applicative f =>
(a -> b -> c) -> f a -> f b -> f c
liftA2 v -> v -> v
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 = [(k, v)] -> v -> SF k v
forall k v. Ord k => [(k, v)] -> v -> SF k v
fromList ([(k, v)] -> v -> SF k v) -> Gen [(k, v)] -> Gen (v -> SF k v)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> Gen [(k, v)]
forall a. Arbitrary a => Gen a
QC.arbitrary Gen (v -> SF k v) -> Gen v -> Gen (SF k v)
forall a b. Gen (a -> b) -> Gen a -> Gen b
forall (f :: * -> *) a b. Applicative f => f (a -> b) -> f a -> f b
<*> Gen v
forall a. Arbitrary a => Gen a
QC.arbitrary
    shrink :: SF k v -> [SF k v]
shrink (SF Map k v
m v
v) = ([(k, v)] -> v -> SF k v) -> ([(k, v)], v) -> SF k v
forall a b c. (a -> b -> c) -> (a, b) -> c
uncurry [(k, v)] -> v -> SF k v
forall k v. Ord k => [(k, v)] -> v -> SF k v
fromList (([(k, v)], v) -> SF k v) -> [([(k, v)], v)] -> [SF k v]
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> ([(k, v)], v) -> [([(k, v)], v)]
forall a. Arbitrary a => a -> [a]
QC.shrink (Map k v -> [(k, v)]
forall k a. Map k a -> [(k, a)]
Map.toList Map k v
m, v
v)

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

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

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

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

-------------------------------------------------------------------------------
-- 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) (k, b) -> [(k, b)] -> [(k, b)]
forall a. a -> [a] -> [a]
: k -> [(k, b)] -> [(k, b)]
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 k -> k -> Bool
forall a. Ord a => a -> a -> Bool
< k
k'    = (k, b)
p (k, b) -> [(k, b)] -> [(k, b)]
forall a. a -> [a] -> [a]
: k -> [(k, b)] -> [(k, b)]
forall k b. Ord k => k -> [(k, b)] -> [(k, b)]
mkDistinctAscList' k
k' [(k, b)]
kv
    | Bool
otherwise =     k -> [(k, b)] -> [(k, b)]
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 k v
m v
def ! :: forall k v. Ord k => SF k v -> k -> v
! k
x = case k -> Map k v -> Maybe (k, v)
forall k v. Ord k => k -> Map k v -> Maybe (k, v)
Map.lookupGT k
x Map k v
m of
    Maybe (k, v)
Nothing     -> v
def
    Just (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 = Map k a -> a -> SF k a
forall k v. Map k v -> v -> SF k v
SF Map k a
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 = Map k v -> v -> SF k v
forall k v. Map k v -> v -> SF k v
SF (Map k v -> v -> SF k v) -> (v -> Map k v) -> v -> v -> SF k v
forall b c a. (b -> c) -> (a -> b) -> a -> c
. k -> v -> Map k v
forall k a. k -> a -> Map k a
Map.singleton k
k

-- | Create function from list of cases and default value.
--
-- >>> putSF $ fromList [(1,2),(3,4)] 5
-- \x -> if
--     | x < 1     -> 2
--     | x < 3     -> 4
--     | otherwise -> 5
--
-- >>> map (fromList [(1,2),(3,4)] 5 !) [0..10]
-- [2,4,4,5,5,5,5,5,5,5,5]
--
fromList :: Ord k => [(k, v)] -> v -> SF k v
fromList :: forall k v. Ord k => [(k, v)] -> v -> SF k v
fromList = Map k v -> v -> SF k v
forall k v. Map k v -> v -> SF k v
SF (Map k v -> v -> SF k v)
-> ([(k, v)] -> Map k v) -> [(k, v)] -> v -> SF k v
forall b c a. (b -> c) -> (a -> b) -> a -> c
. [(k, v)] -> Map k v
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 k v
m v
v) = Map k v -> [v]
forall k a. Map k a -> [a]
Map.elems Map k v
m [v] -> [v] -> [v]
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 k v
m v
v) = ([(k, v)] -> v -> SF k v) -> ([(k, v)], v) -> SF k v
forall a b c. (a -> b -> c) -> (a, b) -> c
uncurry [(k, v)] -> v -> SF k v
forall {k} {p}. [(k, v)] -> p -> SF k v
mk (([(k, v)], v) -> SF k v) -> ([(k, v)], v) -> SF k v
forall a b. (a -> b) -> a -> b
$ ((k, v) -> ([(k, v)], v) -> ([(k, v)], v))
-> ([(k, v)], v) -> [(k, v)] -> ([(k, v)], v)
forall a b. (a -> b -> b) -> b -> [a] -> b
forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr (k, v) -> ([(k, v)], v) -> ([(k, v)], v)
forall {a} {a}. Eq a => (a, a) -> ([(a, a)], a) -> ([(a, a)], a)
go ([], v
v) (Map k v -> [(k, v)]
forall k a. Map k a -> [(k, a)]
Map.toList Map k v
m) where
    mk :: [(k, v)] -> p -> SF k v
mk [(k, v)]
m' p
_ = Map k v -> v -> SF k v
forall k v. Map k v -> v -> SF k v
SF ([(k, v)] -> Map k v
forall k a. [(k, a)] -> Map k a
Map.fromDistinctAscList [(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' a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
x   = ([(a, a)], a)
p'
        | Bool
otherwise = ((a, a)
p (a, a) -> [(a, a)] -> [(a, a)]
forall a. a -> [a] -> [a]
: [(a, a)]
m', a
v')

-------------------------------------------------------------------------------
-- Conversions
-------------------------------------------------------------------------------

-- | Convert from discrete variant to more "dense"
--
-- >>> SF.putSF $ toDense $ fromList [(1,2),(3,4)] 5
-- \x -> if
--     | x < 1     -> 2
--     | x < 3     -> 4
--     | otherwise -> 5
--
toDense :: SF a b -> SF.SF a b
toDense :: forall a b. SF a b -> SF a b
toDense (SF Map a b
m b
v) = Map (Bound a) b -> b -> SF a b
forall k v. Map (Bound k) v -> v -> SF k v
SF.SF ((a -> Bound a) -> Map a b -> Map (Bound a) b
forall k1 k2 a. (k1 -> k2) -> Map k1 a -> Map k2 a
Map.mapKeysMonotonic a -> Bound a
forall k. k -> Bound k
SF.Open Map a b
m) b
v

-- | Convert from "dense" variant. @<= k@ pieces will be converted to @< 'succ' k@.
-- There might be less pieces in the ressult 'SF', than in the original.
--
-- >>> let f = SF.fromList [(SF.Open 1,2),(SF.Closed 3,4),(SF.Open 4,5)] 6
-- >>> SF.putSF f
-- \x -> if
--     | x <  1    -> 2
--     | x <= 3    -> 4
--     | x <  4    -> 5
--     | otherwise -> 6
--
-- >>> putSF $ fromDense (Just . succ) f
-- \x -> if
--     | x < 1     -> 2
--     | x < 4     -> 4
--     | otherwise -> 6
--
fromDense
    :: Ord a
    => (a -> Maybe a) -- ^ next key, if exists
    -> SF.SF a b
    -> SF a b
fromDense :: forall a b. Ord a => (a -> Maybe a) -> SF a b -> SF a b
fromDense a -> Maybe a
next (SF.SF Map (Bound a) b
m b
v) = Map a b -> b -> SF a b
forall k v. Map k v -> v -> SF k v
SF (Map (Bound a) b -> Map a b
forall {a}. Map (Bound a) a -> Map a a
mapKeys Map (Bound a) b
m) b
v where
    mapKeys :: Map (Bound a) a -> Map a a
mapKeys = (a -> a -> a) -> [(a, a)] -> Map a a
forall k a. Ord k => (a -> a -> a) -> [(k, a)] -> Map k a
Map.fromListWith (\a
_ -> a -> a
forall a. a -> a
id) ([(a, a)] -> Map a a)
-> (Map (Bound a) a -> [(a, a)]) -> Map (Bound a) a -> Map a a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. ((Bound a, a) -> Maybe (a, a)) -> [(Bound a, a)] -> [(a, a)]
forall a b. (a -> Maybe b) -> [a] -> [b]
mapMaybe ((Bound a -> Maybe a) -> (Bound a, a) -> Maybe (a, a)
forall (f :: * -> *) a b c.
Functor f =>
(a -> f b) -> (a, c) -> f (b, c)
_1 Bound a -> Maybe a
fk) ([(Bound a, a)] -> [(a, a)])
-> (Map (Bound a) a -> [(Bound a, a)])
-> Map (Bound a) a
-> [(a, a)]
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Map (Bound a) a -> [(Bound a, a)]
forall k a. Map k a -> [(k, a)]
Map.toList

    fk :: Bound a -> Maybe a
fk (SF.Open a
k)   = a -> Maybe a
forall a. a -> Maybe a
Just a
k
    fk (SF.Closed a
k) = a -> Maybe a
next a
k

    _1 :: Functor f => (a -> f b) -> (a, c) -> f (b, c)
    _1 :: forall (f :: * -> *) a b c.
Functor f =>
(a -> f b) -> (a, c) -> f (b, c)
_1 a -> f b
f (a
a, c
c) = (b -> (b, c)) -> f b -> f (b, c)
forall a b. (a -> b) -> f a -> f b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap (\b
b -> (b
b, c
c)) (a -> f b
f a
a)

-------------------------------------------------------------------------------
-- 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 a b
m b
v) | Map a b -> Bool
forall k a. Map k a -> Bool
Map.null Map a b
m = String
"\\_ -> " String -> ShowS
forall a. [a] -> [a] -> [a]
++ b -> String
forall a. Show a => a -> String
show b
v
showSF (SF Map a b
m b
v) = String -> [String] -> String
forall a. [a] -> [[a]] -> [a]
intercalate String
"\n" ([String] -> String) -> [String] -> String
forall a b. (a -> b) -> a -> b
$
    String
"\\x -> if" String -> [String] -> [String]
forall a. a -> [a] -> [a]
: [ String
"    | " String -> ShowS
forall a. [a] -> [a] -> [a]
++ ShowS
leftPad String
k String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
" -> " String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
x | (String
k, String
x) <- [(String, String)]
cases ]
  where
    cases :: [(String, String)]
cases     = [ (String
"x < " String -> ShowS
forall a. [a] -> [a] -> [a]
++ Int -> a -> ShowS
forall a. Show a => Int -> a -> ShowS
showsPrec Int
5 a
k String
"", b -> String
forall a. Show a => a -> String
show b
x) | (a
k,b
x) <- Map a b -> [(a, b)]
forall k a. Map k a -> [(k, a)]
Map.toList Map a b
m ] [(String, String)] -> [(String, String)] -> [(String, String)]
forall a. [a] -> [a] -> [a]
++
                [ (String
"otherwise", b -> String
forall a. Show a => a -> String
show b
v) ]
    len :: Int
len       = [Int] -> Int
forall a. Ord a => [a] -> a
forall (t :: * -> *) a. (Foldable t, Ord a) => t a -> a
maximum (((String, String) -> Int) -> [(String, String)] -> [Int]
forall a b. (a -> b) -> [a] -> [b]
map (String -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length (String -> Int)
-> ((String, String) -> String) -> (String, String) -> Int
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (String, String) -> String
forall a b. (a, b) -> a
fst) [(String, String)]
cases)
    leftPad :: ShowS
leftPad String
s = String
s String -> ShowS
forall a. [a] -> [a] -> [a]
++ Int -> Char -> String
forall a. Int -> a -> [a]
replicate (Int
len Int -> Int -> Int
forall a. Num a => a -> a -> a
- String -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length String
s) Char
' '

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

-- $setup
--
-- >>> import Control.Applicative (liftA2, pure)
-- >>> import qualified Data.Function.Step as SF
-- >>> 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]