{-# LANGUAGE TypeFamilies #-}

-- | Algebra for Actions
--
-- Convention: the |'s in the operators point towards the higher-kinded number, representing an operator or action __into__ a structure.
module NumHask.Algebra.Action
  ( AdditiveAction (..),
    (+|),
    SubtractiveAction (..),
    (-|),
    MultiplicativeAction (..),
    (*|),
    DivisiveAction (..),
    (/|),
    Module,
    TrivialAction (..),
  )
where

import Data.Kind (Type)
import NumHask.Algebra.Additive (Additive (..), Subtractive (..))
import NumHask.Algebra.Multiplicative (Divisive (..), Multiplicative (..))
import NumHask.Algebra.Ring (Distributive)
import Prelude (Double, Eq, Float, Ord, flip)

-- | Additive Action
--
-- > m |+ zero == m
class
  (Additive (AdditiveScalar m)) =>
  AdditiveAction m
  where
  type AdditiveScalar m :: Type

  infixl 6 |+
  (|+) :: m -> AdditiveScalar m -> m

infixl 6 +|

-- | flipped additive action
--
-- > (+|) == flip (|+)
-- > zero +| m = m
(+|) :: (AdditiveAction m) => AdditiveScalar m -> m -> m
+| :: forall m. AdditiveAction m => AdditiveScalar m -> m -> m
(+|) = (m -> AdditiveScalar m -> m) -> AdditiveScalar m -> m -> m
forall a b c. (a -> b -> c) -> b -> a -> c
flip m -> AdditiveScalar m -> m
forall m. AdditiveAction m => m -> AdditiveScalar m -> m
(|+)

-- | Subtractive Action
--
-- > m |- zero = m
class
  (AdditiveAction m, Subtractive (AdditiveScalar m)) =>
  SubtractiveAction m
  where
  infixl 6 |-
  (|-) :: m -> AdditiveScalar m -> m

infixl 6 -|

-- | Subtraction with the scalar on the left
--
-- > (-|) == (+|) . negate
-- > zero -| m = negate m
(-|) :: (AdditiveAction m, Subtractive m) => AdditiveScalar m -> m -> m
AdditiveScalar m
a -| :: forall m.
(AdditiveAction m, Subtractive m) =>
AdditiveScalar m -> m -> m
-| m
b = AdditiveScalar m
a AdditiveScalar m -> m -> m
forall m. AdditiveAction m => AdditiveScalar m -> m -> m
+| m -> m
forall a. Subtractive a => a -> a
negate m
b

-- | Multiplicative Action
--
-- > m |* one = m
-- > m |* zero = zero
class
  (Multiplicative (Scalar m)) =>
  MultiplicativeAction m
  where
  type Scalar m :: Type

  infixl 7 |*
  (|*) :: m -> Scalar m -> m

infixl 7 *|

-- | flipped multiplicative action
--
-- > (*|) == flip (|*)
-- > one *| m = one
-- > zero *| m = zero
(*|) :: (MultiplicativeAction m) => Scalar m -> m -> m
*| :: forall m. MultiplicativeAction m => Scalar m -> m -> m
(*|) = (m -> Scalar m -> m) -> Scalar m -> m -> m
forall a b c. (a -> b -> c) -> b -> a -> c
flip m -> Scalar m -> m
forall m. MultiplicativeAction m => m -> Scalar m -> m
(|*)

-- | Divisive Action
--
-- > m |/ one = m
class
  (Divisive (Scalar m), MultiplicativeAction m) =>
  DivisiveAction m
  where
  infixl 7 |/
  (|/) :: m -> Scalar m -> m

-- | left scalar division
--
-- > (/|) == (*|) . recip
-- > one |/ m = recip m
(/|) :: (MultiplicativeAction m, Divisive m) => Scalar m -> m -> m
Scalar m
a /| :: forall m.
(MultiplicativeAction m, Divisive m) =>
Scalar m -> m -> m
/| m
b = Scalar m
a Scalar m -> m -> m
forall m. MultiplicativeAction m => Scalar m -> m -> m
*| m -> m
forall a. Divisive a => a -> a
recip m
b

-- | A <https://en.wikipedia.org/wiki/Module_(mathematics) Module>
--
-- > a *| one == a
-- > (a + b) *| c == (a *| c) + (b *| c)
-- > c |* (a + b) == (c |* a) + (c |* b)
-- > a *| zero == zero
-- > a *| b == b |* a
type Module m = (Distributive (Scalar m), MultiplicativeAction m)

-- | Scalar self-action for 'Double'.
--
-- This used to live as an orphan in @circuits-stats@ ('Circuit.Stats.ODE')
-- because it was missing from the numhask class hierarchy. Moving it here
-- removes the orphan and makes the action available to any downstream package.
instance MultiplicativeAction Double where
  type Scalar Double = Double
  |* :: Double -> Scalar Double -> Double
(|*) = Double -> Double -> Double
Double -> Scalar Double -> Double
forall a. Multiplicative a => a -> a -> a
(*)

instance DivisiveAction Double where
  |/ :: Double -> Scalar Double -> Double
(|/) = Double -> Double -> Double
Double -> Scalar Double -> Double
forall a. Divisive a => a -> a -> a
(/)

-- | Scalar self-action for 'Float'.
instance MultiplicativeAction Float where
  type Scalar Float = Float
  |* :: Float -> Scalar Float -> Float
(|*) = Float -> Float -> Float
Float -> Scalar Float -> Float
forall a. Multiplicative a => a -> a -> a
(*)

instance DivisiveAction Float where
  |/ :: Float -> Scalar Float -> Float
(|/) = Float -> Float -> Float
Float -> Scalar Float -> Float
forall a. Divisive a => a -> a -> a
(/)

-- | An action of a set of numbers on itself
newtype TrivialAction a = TrivialAction
  { forall a. TrivialAction a -> a
getTrivialAction :: a
  }
  deriving (TrivialAction a -> TrivialAction a -> Bool
(TrivialAction a -> TrivialAction a -> Bool)
-> (TrivialAction a -> TrivialAction a -> Bool)
-> Eq (TrivialAction a)
forall a. Eq a => TrivialAction a -> TrivialAction a -> Bool
forall a. (a -> a -> Bool) -> (a -> a -> Bool) -> Eq a
$c== :: forall a. Eq a => TrivialAction a -> TrivialAction a -> Bool
== :: TrivialAction a -> TrivialAction a -> Bool
$c/= :: forall a. Eq a => TrivialAction a -> TrivialAction a -> Bool
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Eq, Eq (TrivialAction a)
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(TrivialAction a -> TrivialAction a -> Ordering)
-> (TrivialAction a -> TrivialAction a -> Bool)
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-> (TrivialAction a -> TrivialAction a -> Bool)
-> (TrivialAction a -> TrivialAction a -> TrivialAction a)
-> (TrivialAction a -> TrivialAction a -> TrivialAction a)
-> Ord (TrivialAction a)
TrivialAction a -> TrivialAction a -> Bool
TrivialAction a -> TrivialAction a -> Ordering
TrivialAction a -> TrivialAction a -> TrivialAction a
forall a.
Eq a =>
(a -> a -> Ordering)
-> (a -> a -> Bool)
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forall a. Ord a => Eq (TrivialAction a)
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TrivialAction a -> TrivialAction a -> TrivialAction a
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(TrivialAction a -> TrivialAction a -> TrivialAction a)
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forall a. Additive a => TrivialAction a
forall a.
Additive a =>
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forall a. (a -> a -> a) -> a -> Additive a
$c+ :: forall a.
Additive a =>
TrivialAction a -> TrivialAction a -> TrivialAction a
+ :: TrivialAction a -> TrivialAction a -> TrivialAction a
$czero :: forall a. Additive a => TrivialAction a
zero :: TrivialAction a
Additive, Additive (TrivialAction a)
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-> Subtractive (TrivialAction a)
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TrivialAction a -> TrivialAction a -> TrivialAction a
forall a. Additive a => (a -> a) -> (a -> a -> a) -> Subtractive a
forall a. Subtractive a => Additive (TrivialAction a)
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$cnegate :: forall a. Subtractive a => TrivialAction a -> TrivialAction a
negate :: TrivialAction a -> TrivialAction a
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Subtractive a =>
TrivialAction a -> TrivialAction a -> TrivialAction a
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Divisive a =>
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/ :: TrivialAction a -> TrivialAction a -> TrivialAction a
Divisive)

instance (Additive a) => AdditiveAction (TrivialAction a) where
  type AdditiveScalar (TrivialAction a) = a
  TrivialAction a
a |+ :: TrivialAction a
-> AdditiveScalar (TrivialAction a) -> TrivialAction a
|+ AdditiveScalar (TrivialAction a)
b = a -> TrivialAction a
forall a. a -> TrivialAction a
TrivialAction (a
a a -> a -> a
forall a. Additive a => a -> a -> a
+ a
AdditiveScalar (TrivialAction a)
b)

instance (Subtractive a) => SubtractiveAction (TrivialAction a) where
  TrivialAction a
a |- :: TrivialAction a
-> AdditiveScalar (TrivialAction a) -> TrivialAction a
|- AdditiveScalar (TrivialAction a)
b = a -> TrivialAction a
forall a. a -> TrivialAction a
TrivialAction (a
a a -> a -> a
forall a. Subtractive a => a -> a -> a
- a
AdditiveScalar (TrivialAction a)
b)

instance (Multiplicative a) => MultiplicativeAction (TrivialAction a) where
  type Scalar (TrivialAction a) = a
  TrivialAction a
a |* :: TrivialAction a -> Scalar (TrivialAction a) -> TrivialAction a
|* Scalar (TrivialAction a)
b = a -> TrivialAction a
forall a. a -> TrivialAction a
TrivialAction (a
a a -> a -> a
forall a. Multiplicative a => a -> a -> a
* a
Scalar (TrivialAction a)
b)

instance (Divisive a) => DivisiveAction (TrivialAction a) where
  TrivialAction a
a |/ :: TrivialAction a -> Scalar (TrivialAction a) -> TrivialAction a
|/ Scalar (TrivialAction a)
b = a -> TrivialAction a
forall a. a -> TrivialAction a
TrivialAction (a
a a -> a -> a
forall a. Divisive a => a -> a -> a
/ a
Scalar (TrivialAction a)
b)