-- Hoogle documentation, generated by Haddock
-- See Hoogle, http://www.haskell.org/hoogle/
-- | Staircase functions or piecewise constant functions
--
-- Step functions, staircase functions or piecewise constant functions.
-- Implemented as a default value and a series of transitions. Supports
-- merging two step functions using a supplied merging function.
@package step-function
@version 0.2
module Data.Function.Step
-- | 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
-- <math>:
--
-- <math>
--
-- or as you would write in Haskell
--
--
-- f x | x < k1 = v1
-- | x <= k2 = v2
-- | otherwise = v3
--
--
-- Note: total-map 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).
--
--
-- liftA2 f g h ! x == liftA2 f (g !) (h !) x
--
--
-- Recall that for ordinary functions liftA2 f g h x = f (g x)
-- (h x).
--
-- Dense?
--
-- This dense variant is useful with dense ordered 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 -> SF k v
-- | Bound operations
data Bound k
-- | less-than, <
Open :: k -> Bound k
-- | less-than-or-equal, ≤.
Closed :: k -> Bound k
-- | Constant function
--
--
-- >>> putSF $ constant 1
-- \_ -> 1
--
constant :: a -> SF k a
-- | 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
-- | 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
-- | 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
--
--
--
-- normalise (liftA2 (+) p (fmap negate p)) == (pure 0 :: SF Int Int)
--
normalise :: Eq v => SF k v -> SF k v
-- | Apply SF.
--
--
-- >>> heaviside ! 2
-- 1
--
(!) :: Ord k => SF k v -> k -> v
infixl 9 !
-- | Possible values of SF
--
--
-- >>> values heaviside
-- [-1,1]
--
values :: SF k v -> [v]
-- | Show SF as Haskell code
showSF :: (Show a, Show b) => SF a b -> String
-- |
-- putStrLn . showSF
--
putSF :: (Show a, Show b) => SF a b -> IO ()
instance Data.Traversable.Traversable (Data.Function.Step.SF k)
instance Data.Foldable.Foldable (Data.Function.Step.SF k)
instance GHC.Base.Functor (Data.Function.Step.SF k)
instance (GHC.Classes.Ord v, GHC.Classes.Ord k) => GHC.Classes.Ord (Data.Function.Step.SF k v)
instance (GHC.Classes.Eq v, GHC.Classes.Eq k) => GHC.Classes.Eq (Data.Function.Step.SF k v)
instance Data.Traversable.Traversable Data.Function.Step.Bound
instance Data.Foldable.Foldable Data.Function.Step.Bound
instance GHC.Base.Functor Data.Function.Step.Bound
instance GHC.Show.Show k => GHC.Show.Show (Data.Function.Step.Bound k)
instance GHC.Classes.Eq k => GHC.Classes.Eq (Data.Function.Step.Bound k)
instance GHC.Classes.Ord k => GHC.Base.Applicative (Data.Function.Step.SF k)
instance GHC.Classes.Ord k => GHC.Base.Monad (Data.Function.Step.SF k)
instance (GHC.Classes.Ord k, Data.Semigroup.Semigroup v) => Data.Semigroup.Semigroup (Data.Function.Step.SF k v)
instance (GHC.Classes.Ord k, GHC.Base.Monoid v) => GHC.Base.Monoid (Data.Function.Step.SF k v)
instance (GHC.Classes.Ord k, Test.QuickCheck.Arbitrary.Arbitrary k, Test.QuickCheck.Arbitrary.Arbitrary v) => Test.QuickCheck.Arbitrary.Arbitrary (Data.Function.Step.SF k v)
instance (Control.DeepSeq.NFData k, Control.DeepSeq.NFData v) => Control.DeepSeq.NFData (Data.Function.Step.SF k v)
instance Data.Functor.Classes.Show2 Data.Function.Step.SF
instance GHC.Show.Show k => Data.Functor.Classes.Show1 (Data.Function.Step.SF k)
instance (GHC.Show.Show k, GHC.Show.Show v) => GHC.Show.Show (Data.Function.Step.SF k v)
instance GHC.Classes.Ord k => GHC.Classes.Ord (Data.Function.Step.Bound k)
instance Test.QuickCheck.Arbitrary.Arbitrary k => Test.QuickCheck.Arbitrary.Arbitrary (Data.Function.Step.Bound k)
instance Control.DeepSeq.NFData k => Control.DeepSeq.NFData (Data.Function.Step.Bound k)
instance Data.Functor.Classes.Show1 Data.Function.Step.Bound
module Data.Function.Step.Discrete.Closed
-- | 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 <math>:
--
-- <math>
--
-- 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 -> SF k v
-- | Constant function
--
--
-- >>> putSF $ constant 1
-- \_ -> 1
--
constant :: a -> SF k a
-- | 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
-- | Create function from list of cases and default value.
--
--
-- >>> let f = fromList [(1,2),(3,4)] 5
--
-- >>> putSF f
-- \x -> if
-- | x <= 1 -> 2
-- | x <= 3 -> 4
-- | otherwise -> 5
--
--
--
-- >>> map (f !) [0..10]
-- [2,2,4,4,5,5,5,5,5,5,5]
--
fromList :: Ord k => [(k, v)] -> v -> SF k v
-- | 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
--
--
--
-- normalise (liftA2 (+) p (fmap negate p)) == (pure 0 :: SF Int Int)
--
normalise :: Eq v => SF k v -> SF k v
-- | Apply SF.
--
--
-- >>> heaviside ! 2
-- 1
--
(!) :: Ord k => SF k v -> k -> v
infixl 9 !
-- | Possible values of SF
--
--
-- >>> values heaviside
-- [-1,1]
--
values :: SF k v -> [v]
-- | 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 a b
-- | 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 . pred) f
-- \x -> if
-- | x <= 0 -> 2
-- | x <= 3 -> 4
-- | otherwise -> 6
--
fromDense :: Ord a => (a -> Maybe a) -> SF a b -> SF a b
-- | Show SF as Haskell code
showSF :: (Show a, Show b) => SF a b -> String
-- |
-- putStrLn . showSF
--
putSF :: (Show a, Show b) => SF a b -> IO ()
instance Data.Traversable.Traversable (Data.Function.Step.Discrete.Closed.SF k)
instance Data.Foldable.Foldable (Data.Function.Step.Discrete.Closed.SF k)
instance GHC.Base.Functor (Data.Function.Step.Discrete.Closed.SF k)
instance (GHC.Classes.Ord v, GHC.Classes.Ord k) => GHC.Classes.Ord (Data.Function.Step.Discrete.Closed.SF k v)
instance (GHC.Classes.Eq v, GHC.Classes.Eq k) => GHC.Classes.Eq (Data.Function.Step.Discrete.Closed.SF k v)
instance GHC.Classes.Ord k => GHC.Base.Applicative (Data.Function.Step.Discrete.Closed.SF k)
instance GHC.Classes.Ord k => GHC.Base.Monad (Data.Function.Step.Discrete.Closed.SF k)
instance (GHC.Classes.Ord k, Data.Semigroup.Semigroup v) => Data.Semigroup.Semigroup (Data.Function.Step.Discrete.Closed.SF k v)
instance (GHC.Classes.Ord k, GHC.Base.Monoid v) => GHC.Base.Monoid (Data.Function.Step.Discrete.Closed.SF k v)
instance (GHC.Classes.Ord k, Test.QuickCheck.Arbitrary.Arbitrary k, Test.QuickCheck.Arbitrary.Arbitrary v) => Test.QuickCheck.Arbitrary.Arbitrary (Data.Function.Step.Discrete.Closed.SF k v)
instance (Control.DeepSeq.NFData k, Control.DeepSeq.NFData v) => Control.DeepSeq.NFData (Data.Function.Step.Discrete.Closed.SF k v)
instance Data.Functor.Classes.Show2 Data.Function.Step.Discrete.Closed.SF
instance GHC.Show.Show k => Data.Functor.Classes.Show1 (Data.Function.Step.Discrete.Closed.SF k)
instance (GHC.Show.Show k, GHC.Show.Show v) => GHC.Show.Show (Data.Function.Step.Discrete.Closed.SF k v)
module Data.Function.Step.Discrete.Open
-- | 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 <math>:
--
-- <math>
--
-- 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 -> SF k v
-- | Constant function
--
--
-- >>> putSF $ constant 1
-- \_ -> 1
--
constant :: a -> SF k a
-- | 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
-- | 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
-- | 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
--
--
--
-- normalise (liftA2 (+) p (fmap negate p)) == (pure 0 :: SF Int Int)
--
normalise :: Eq v => SF k v -> SF k v
-- | Apply SF.
--
--
-- >>> heaviside ! 2
-- 1
--
(!) :: Ord k => SF k v -> k -> v
infixl 9 !
-- | Possible values of SF
--
--
-- >>> values heaviside
-- [-1,1]
--
values :: SF k v -> [v]
-- | 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 a b
-- | 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) -> SF a b -> SF a b
-- | Show SF as Haskell code
showSF :: (Show a, Show b) => SF a b -> String
-- |
-- putStrLn . showSF
--
putSF :: (Show a, Show b) => SF a b -> IO ()
instance Data.Traversable.Traversable (Data.Function.Step.Discrete.Open.SF k)
instance Data.Foldable.Foldable (Data.Function.Step.Discrete.Open.SF k)
instance GHC.Base.Functor (Data.Function.Step.Discrete.Open.SF k)
instance (GHC.Classes.Ord v, GHC.Classes.Ord k) => GHC.Classes.Ord (Data.Function.Step.Discrete.Open.SF k v)
instance (GHC.Classes.Eq v, GHC.Classes.Eq k) => GHC.Classes.Eq (Data.Function.Step.Discrete.Open.SF k v)
instance GHC.Classes.Ord k => GHC.Base.Applicative (Data.Function.Step.Discrete.Open.SF k)
instance GHC.Classes.Ord k => GHC.Base.Monad (Data.Function.Step.Discrete.Open.SF k)
instance (GHC.Classes.Ord k, Data.Semigroup.Semigroup v) => Data.Semigroup.Semigroup (Data.Function.Step.Discrete.Open.SF k v)
instance (GHC.Classes.Ord k, GHC.Base.Monoid v) => GHC.Base.Monoid (Data.Function.Step.Discrete.Open.SF k v)
instance (GHC.Classes.Ord k, Test.QuickCheck.Arbitrary.Arbitrary k, Test.QuickCheck.Arbitrary.Arbitrary v) => Test.QuickCheck.Arbitrary.Arbitrary (Data.Function.Step.Discrete.Open.SF k v)
instance (Control.DeepSeq.NFData k, Control.DeepSeq.NFData v) => Control.DeepSeq.NFData (Data.Function.Step.Discrete.Open.SF k v)
instance Data.Functor.Classes.Show2 Data.Function.Step.Discrete.Open.SF
instance GHC.Show.Show k => Data.Functor.Classes.Show1 (Data.Function.Step.Discrete.Open.SF k)
instance (GHC.Show.Show k, GHC.Show.Show v) => GHC.Show.Show (Data.Function.Step.Discrete.Open.SF k v)
-- | This module re-exports Function.Step.Discrete.Open
module Data.Function.Step.Discrete