{-# OPTIONS_GHC -fno-warn-orphans #-} module Pandora.Paradigm.Primary.Algebraic.Exponential where import Pandora.Pattern.Betwixt (Betwixt) import Pandora.Pattern.Semigroupoid (Semigroupoid ((.))) import Pandora.Pattern.Category (Category ((#), identity)) import Pandora.Pattern.Kernel (Kernel (constant)) import Pandora.Pattern.Functor.Covariant (Covariant ((<-|-))) import Pandora.Pattern.Functor.Contravariant (Contravariant ((>-|-))) import Pandora.Pattern.Functor.Distributive (Distributive ((-<<))) import Pandora.Pattern.Functor.Bindable (Bindable ((=<<))) import Pandora.Pattern.Object.Semigroup (Semigroup ((+))) import Pandora.Pattern.Object.Ringoid (Ringoid ((*))) import Pandora.Pattern.Morphism.Flip (Flip (Flip)) import Pandora.Pattern.Morphism.Straight (Straight (Straight)) infixr 7 .:.. infixr 9 % infixl 1 & type instance Betwixt (->) (->) = (->) instance Semigroupoid (->) where b -> c f . :: (b -> c) -> (a -> b) -> a -> c . a -> b g = \a x -> b -> c f (a -> b g a x) instance Category (->) where identity :: a -> a identity a x = a x instance Kernel (->) where constant :: a -> i -> a constant a x i _ = a x instance Covariant (->) (->) ((->) a) where <-|- :: (a -> b) -> (a -> a) -> a -> b (<-|-) = (a -> b) -> (a -> a) -> a -> b forall (m :: * -> * -> *) b c a. Semigroupoid m => m b c -> m a b -> m a c (.) instance Distributive (->) (->) ((->) e) where a -> e -> b f -<< :: (a -> e -> b) -> u a -> e -> u b -<< u a g = \e e -> (a -> e -> b f (a -> e -> b) -> e -> a -> b forall a b c. (a -> b -> c) -> b -> a -> c % e e) (a -> b) -> u a -> u b forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => source a b -> target (t a) (t b) <-|- u a g instance Bindable (->) ((->) e) where a -> e -> b f =<< :: (a -> e -> b) -> (e -> a) -> e -> b =<< e -> a g = \e x -> a -> e -> b f (a -> e -> b) -> a -> e -> b forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) # e -> a g e x (e -> b) -> e -> b forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b) # e x instance Semigroup r => Semigroup (e -> r) where e -> r f + :: (e -> r) -> (e -> r) -> e -> r + e -> r g = \e e -> e -> r f e e r -> r -> r forall a. Semigroup a => a -> a -> a + e -> r g e e instance Ringoid r => Ringoid (e -> r) where e -> r f * :: (e -> r) -> (e -> r) -> e -> r * e -> r g = \e e -> e -> r f e e r -> r -> r forall a. Ringoid a => a -> a -> a * e -> r g e e type (<--) = Flip (->) instance Contravariant (->) (->) ((<--) a) where a -> b f >-|- :: (a -> b) -> (a <-- b) -> a <-- a >-|- Flip b -> a g = (a -> a) -> a <-- a forall (v :: * -> * -> *) a e. v e a -> Flip v a e Flip (b -> a g (b -> a) -> (a -> b) -> a -> a forall (m :: * -> * -> *) b c a. Semigroupoid m => m b c -> m a b -> m a c . a -> b f) type (-->) = Straight (->) instance Covariant (->) (->) ((-->) b) where a -> b f <-|- :: (a -> b) -> (b --> a) -> b --> b <-|- Straight b -> a g = (b -> b) -> b --> b forall (v :: * -> * -> *) a e. v a e -> Straight v a e Straight (a -> b f (a -> b) -> (b -> a) -> b -> b forall (m :: * -> * -> *) b c a. Semigroupoid m => m b c -> m a b -> m a c . b -> a g) (.:..) :: (Covariant (->) target (v a), Semigroupoid v) => v c d -> target (v a (v b c)) (v a (v b d)) .:.. :: v c d -> target (v a (v b c)) (v a (v b d)) (.:..) v c d f = (v b c -> v b d) -> target (v a (v b c)) (v a (v b d)) forall (source :: * -> * -> *) (target :: * -> * -> *) (t :: * -> *) a b. Covariant source target t => source a b -> target (t a) (t b) (<-|-) (v c d f v c d -> v b c -> v b d forall (m :: * -> * -> *) b c a. Semigroupoid m => m b c -> m a b -> m a c .) {-# INLINE (%) #-} (%) :: (a -> b -> c) -> b -> a -> c % :: (a -> b -> c) -> b -> a -> c (%) a -> b -> c f b x a y = a -> b -> c f a y b x {-# INLINE (&) #-} (&) :: a -> (a -> b) -> b a x & :: a -> (a -> b) -> b & a -> b f = a -> b f a x fix :: (a -> a) -> a fix :: (a -> a) -> a fix a -> a f = let x :: a x = a -> a f a x in a x