Algebra.Types

Documentation

class SumKind k

Associated Types

data (a :: k) ⊕ (b :: k) :: k

data Zero :: k

Instances

SumKind Type

Defined in Algebra.Types

Associated Types

data a ⊕ b :: k

data Zero :: k

SumKind (Type -> Type)

Defined in Algebra.Types

Associated Types

data a ⊕ b :: k

data Zero :: k

class ProdKind k

Associated Types

data (a :: k) ⊗ (b :: k) :: k

data One :: k

Instances

ProdKind Type

Defined in Algebra.Types

Associated Types

data a ⊗ b :: k

data One :: k

ProdKind (Type -> Type)

Defined in Algebra.Types

Associated Types

data a ⊗ b :: k

data One :: k

class DualKind k

Associated Types

data Dual (a :: k) :: k

Instances

DualKind Type

Defined in Algebra.Types

Associated Types

data Dual a :: k

DualKind (Type -> Type)

Defined in Algebra.Types

Associated Types

data Dual a :: k

data Repr x i t o :: k -> Type where

Constructors

RPlus :: Repr x i t o a -> Repr x i t o b -> Repr x i t o (a `t` b)

RTimes :: Repr x i t o a -> Repr x i t o b -> Repr x i t o (a `x` b)

ROne :: Repr x i t o i

RZero :: Repr x i t o o

Instances

Arbitrary (Some1 (Repr x i t o))

Defined in Algebra.Category.Objects

Methods

arbitrary :: Gen (Some1 (Repr x i t o))

shrink :: Some1 (Repr x i t o) -> [Some1 (Repr x i t o)]

Show (Repr x i t o a)

Defined in Algebra.Types

Methods

showsPrec :: Int -> Repr x i t o a -> ShowS

show :: Repr x i t o a -> String

showList :: [Repr x i t o a] -> ShowS

type CRepr = Repr (∘) Id (⊗) One

type MRepr = Repr (⊗) One (⊕) Zero

inhabitants :: Finite a => [a]

class (Enum a, Bounded a, Eq a, Ord a) => Finite a where

Minimal complete definition

Nothing

Methods

typeSize :: Int

finiteFstsnd :: forall α β. a ~ (α ⊗ β) => Dict (Finite α, Finite β)

finiteLeftRight :: forall α β. a ~ (α ⊕ β) => Dict (Finite α, Finite β)

Instances

Finite Bool

Defined in Algebra.Types

Methods

typeSize :: Int

finiteFstsnd :: Bool ~ (α ⊗ β) => Dict (Finite α, Finite β)

finiteLeftRight :: Bool ~ (α ⊕ β) => Dict (Finite α, Finite β)

Finite (One :: Type)

Defined in Algebra.Types

Methods

typeSize :: Int

finiteFstsnd :: One ~ (α ⊗ β) => Dict (Finite α, Finite β)

finiteLeftRight :: One ~ (α ⊕ β) => Dict (Finite α, Finite β)

Finite (Zero :: Type)

Defined in Algebra.Types

Methods

typeSize :: Int

finiteFstsnd :: Zero ~ (α ⊗ β) => Dict (Finite α, Finite β)

finiteLeftRight :: Zero ~ (α ⊕ β) => Dict (Finite α, Finite β)

Finite a => Finite (Dual a)

Defined in Algebra.Types

Methods

typeSize :: Int

finiteFstsnd :: Dual a ~ (α ⊗ β) => Dict (Finite α, Finite β)

finiteLeftRight :: Dual a ~ (α ⊕ β) => Dict (Finite α, Finite β)

(Finite x, Finite y) => Finite (x ⊕ y)

Defined in Algebra.Types

Methods

typeSize :: Int

finiteFstsnd :: (x ⊕ y) ~ (α ⊗ β) => Dict (Finite α, Finite β)

finiteLeftRight :: (x ⊕ y) ~ (α ⊕ β) => Dict (Finite α, Finite β)

(Finite x, Finite y) => Finite (x ⊗ y)

Defined in Algebra.Types

Methods

typeSize :: Int

finiteFstsnd :: (x ⊗ y) ~ (α ⊗ β) => Dict (Finite α, Finite β)

finiteLeftRight :: (x ⊗ y) ~ (α ⊕ β) => Dict (Finite α, Finite β)

fromZero :: forall a. Finite a => Int -> a

newtype (f ∘ g) x

Constructors

Comp

Fields

• fromComp :: f (g x)

Instances

Functor f => Generic1 (f ∘ g :: k -> Type)

Defined in Algebra.Types

Associated Types

type Rep1 (f ∘ g) :: k -> Type

Methods

from1 :: forall (a :: k0). (f ∘ g) a -> Rep1 (f ∘ g) a

to1 :: forall (a :: k0). Rep1 (f ∘ g) a -> (f ∘ g) a

Monoidal ((∘) :: (Type -> Type) -> (Type -> Type) -> Type -> Type) Id NatTrans

Defined in Algebra.Category.NatTrans

Methods

(⊗) :: forall (a :: k) (b :: k) (c :: k) (d :: k). (Obj NatTrans a, Obj NatTrans b, Obj NatTrans c, Obj NatTrans d) => NatTrans a b -> NatTrans c d -> NatTrans (a ∘ c) (b ∘ d)

assoc :: forall (a :: k) (b :: k) (c :: k). (Obj NatTrans a, Obj NatTrans b, Obj NatTrans c) => NatTrans ((a ∘ b) ∘ c) (a ∘ (b ∘ c))

assoc_ :: forall (a :: k) (b :: k) (c :: k). (Obj NatTrans a, Obj NatTrans b, Obj NatTrans c) => NatTrans (a ∘ (b ∘ c)) ((a ∘ b) ∘ c)

unitorR :: forall (a :: k). (Obj NatTrans a, Obj NatTrans Id) => NatTrans a (a ∘ Id)

unitorR_ :: forall (a :: k). (Obj NatTrans a, Obj NatTrans Id) => NatTrans (a ∘ Id) a

unitorL :: forall (a :: k). (Obj NatTrans a, Obj NatTrans Id) => NatTrans a (Id ∘ a)

unitorL_ :: forall (a :: k). (Obj NatTrans a, Obj NatTrans Id) => NatTrans (Id ∘ a) a

Ring s => Braided ((∘) :: (Type -> Type) -> (Type -> Type) -> Type -> Type) Id (Mat s :: (Type -> Type) -> (Type -> Type) -> Type)

Defined in Algebra.Linear

Methods

swap :: forall (a :: k) (b :: k). (Obj (Mat s) a, Obj (Mat s) b) => Mat s (a ∘ b) (b ∘ a)

swap_ :: forall (a :: k) (b :: k). (Obj (Mat s) a, Obj (Mat s) b) => Mat s (a ∘ b) (b ∘ a)

Ring s => Monoidal ((∘) :: (Type -> Type) -> (Type -> Type) -> Type -> Type) Id (Mat s :: (Type -> Type) -> (Type -> Type) -> Type)

Defined in Algebra.Linear

Methods

(⊗) :: forall (a :: k) (b :: k) (c :: k) (d :: k). (Obj (Mat s) a, Obj (Mat s) b, Obj (Mat s) c, Obj (Mat s) d) => Mat s a b -> Mat s c d -> Mat s (a ∘ c) (b ∘ d)

assoc :: forall (a :: k) (b :: k) (c :: k). (Obj (Mat s) a, Obj (Mat s) b, Obj (Mat s) c) => Mat s ((a ∘ b) ∘ c) (a ∘ (b ∘ c))

assoc_ :: forall (a :: k) (b :: k) (c :: k). (Obj (Mat s) a, Obj (Mat s) b, Obj (Mat s) c) => Mat s (a ∘ (b ∘ c)) ((a ∘ b) ∘ c)

unitorR :: forall (a :: k). (Obj (Mat s) a, Obj (Mat s) Id) => Mat s a (a ∘ Id)

unitorR_ :: forall (a :: k). (Obj (Mat s) a, Obj (Mat s) Id) => Mat s (a ∘ Id) a

unitorL :: forall (a :: k). (Obj (Mat s) a, Obj (Mat s) Id) => Mat s a (Id ∘ a)

unitorL_ :: forall (a :: k). (Obj (Mat s) a, Obj (Mat s) Id) => Mat s (Id ∘ a) a

Ring s => Symmetric ((∘) :: (Type -> Type) -> (Type -> Type) -> Type -> Type) Id (Mat s :: (Type -> Type) -> (Type -> Type) -> Type)

Defined in Algebra.Linear

(Arbitrary1 f, Arbitrary1 g) => Arbitrary1 (f ∘ g)

Defined in Algebra.Types

Methods

liftArbitrary :: Gen a -> Gen ((f ∘ g) a)

liftShrink :: (a -> [a]) -> (f ∘ g) a -> [(f ∘ g) a]

(Representable v, Representable w) => Representable (v ∘ w)

Defined in Algebra.Types

Associated Types

type Rep (v ∘ w)

Methods

tabulate :: (Rep (v ∘ w) -> a) -> (v ∘ w) a

index :: (v ∘ w) a -> Rep (v ∘ w) -> a

(Foldable f, Foldable g) => Foldable (f ∘ g)

Defined in Algebra.Types

Methods

fold :: Monoid m => (f ∘ g) m -> m

foldMap :: Monoid m => (a -> m) -> (f ∘ g) a -> m

foldMap' :: Monoid m => (a -> m) -> (f ∘ g) a -> m

foldr :: (a -> b -> b) -> b -> (f ∘ g) a -> b

foldr' :: (a -> b -> b) -> b -> (f ∘ g) a -> b

foldl :: (b -> a -> b) -> b -> (f ∘ g) a -> b

foldl' :: (b -> a -> b) -> b -> (f ∘ g) a -> b

foldr1 :: (a -> a -> a) -> (f ∘ g) a -> a

foldl1 :: (a -> a -> a) -> (f ∘ g) a -> a

toList :: (f ∘ g) a -> [a]

null :: (f ∘ g) a -> Bool

length :: (f ∘ g) a -> Int

elem :: Eq a => a -> (f ∘ g) a -> Bool

maximum :: Ord a => (f ∘ g) a -> a

minimum :: Ord a => (f ∘ g) a -> a

sum :: Num a => (f ∘ g) a -> a

product :: Num a => (f ∘ g) a -> a

(Traversable f, Traversable g) => Traversable (f ∘ g)

Defined in Algebra.Types

Methods

traverse :: Applicative f0 => (a -> f0 b) -> (f ∘ g) a -> f0 ((f ∘ g) b)

sequenceA :: Applicative f0 => (f ∘ g) (f0 a) -> f0 ((f ∘ g) a)

mapM :: Monad m => (a -> m b) -> (f ∘ g) a -> m ((f ∘ g) b)

sequence :: Monad m => (f ∘ g) (m a) -> m ((f ∘ g) a)

(Applicative f, Applicative g) => Applicative (f ∘ g)

Defined in Algebra.Types

Methods

pure :: a -> (f ∘ g) a

(<*>) :: (f ∘ g) (a -> b) -> (f ∘ g) a -> (f ∘ g) b

liftA2 :: (a -> b -> c) -> (f ∘ g) a -> (f ∘ g) b -> (f ∘ g) c

(*>) :: (f ∘ g) a -> (f ∘ g) b -> (f ∘ g) b

(<*) :: (f ∘ g) a -> (f ∘ g) b -> (f ∘ g) a

(Functor f, Functor g) => Functor (f ∘ g)

Defined in Algebra.Types

Methods

fmap :: (a -> b) -> (f ∘ g) a -> (f ∘ g) b

(<$) :: a -> (f ∘ g) b -> (f ∘ g) a

(Distributive v, Distributive w) => Distributive (v ∘ w)

Defined in Algebra.Types

Methods

distribute :: Functor f => f ((v ∘ w) a) -> (v ∘ w) (f a)

collect :: Functor f => (a -> (v ∘ w) b) -> f a -> (v ∘ w) (f b)

distributeM :: Monad m => m ((v ∘ w) a) -> (v ∘ w) (m a)

collectM :: Monad m => (a -> (v ∘ w) b) -> m a -> (v ∘ w) (m b)

(VectorR v, VectorR w) => VectorR (v ∘ w)

Defined in Algebra.Linear

Methods

vectorSplit :: forall (f :: Type -> Type) (g :: Type -> Type). (v ∘ w) ~ (f ⊗ g) => Dict (VectorR f, VectorR g)

vectorCut :: forall (f :: Type -> Type) (g :: Type -> Type). (v ∘ w) ~ (f ⊕ g) => Dict (VectorR f, VectorR g)

Show (a (b x)) => Show ((a ∘ b) x)

Defined in Algebra.Types

Methods

showsPrec :: Int -> (a ∘ b) x -> ShowS

show :: (a ∘ b) x -> String

showList :: [(a ∘ b) x] -> ShowS

Eq (f (g x)) => Eq ((f ∘ g) x)

Defined in Algebra.Types

Methods

(==) :: (f ∘ g) x -> (f ∘ g) x -> Bool

(/=) :: (f ∘ g) x -> (f ∘ g) x -> Bool

type Rep1 (f ∘ g :: k -> Type)

Defined in Algebra.Types

type Rep1 (f ∘ g :: k -> Type) = D1 ('MetaData "\8728" "Algebra.Types" "gasp-1.4.0.0-4Z31dYmTpJbCzuuYkBBWRm" 'True) (C1 ('MetaCons "Comp" 'PrefixI 'True) (S1 ('MetaSel ('Just "fromComp") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (f :.: Rec1 g)))

type Rep (v ∘ w)

Defined in Algebra.Types

type Rep (v ∘ w) = Rep v ⊗ Rep w

newtype Id x

Constructors

Id

Fields

• fromId :: x

Instances

Arbitrary1 Id

Defined in Algebra.Types

Methods

liftArbitrary :: Gen a -> Gen (Id a)

liftShrink :: (a -> [a]) -> Id a -> [Id a]

Representable Id

Defined in Algebra.Types

Associated Types

type Rep Id

Methods

tabulate :: (Rep Id -> a) -> Id a

index :: Id a -> Rep Id -> a

Foldable Id

Defined in Algebra.Types

Methods

fold :: Monoid m => Id m -> m

foldMap :: Monoid m => (a -> m) -> Id a -> m

foldMap' :: Monoid m => (a -> m) -> Id a -> m

foldr :: (a -> b -> b) -> b -> Id a -> b

foldr' :: (a -> b -> b) -> b -> Id a -> b

foldl :: (b -> a -> b) -> b -> Id a -> b

foldl' :: (b -> a -> b) -> b -> Id a -> b

foldr1 :: (a -> a -> a) -> Id a -> a

foldl1 :: (a -> a -> a) -> Id a -> a

toList :: Id a -> [a]

null :: Id a -> Bool

length :: Id a -> Int

elem :: Eq a => a -> Id a -> Bool

maximum :: Ord a => Id a -> a

minimum :: Ord a => Id a -> a

sum :: Num a => Id a -> a

product :: Num a => Id a -> a

Traversable Id

Defined in Algebra.Types

Methods

traverse :: Applicative f => (a -> f b) -> Id a -> f (Id b)

sequenceA :: Applicative f => Id (f a) -> f (Id a)

mapM :: Monad m => (a -> m b) -> Id a -> m (Id b)

sequence :: Monad m => Id (m a) -> m (Id a)

Applicative Id

Defined in Algebra.Types

Methods

pure :: a -> Id a

(<*>) :: Id (a -> b) -> Id a -> Id b

liftA2 :: (a -> b -> c) -> Id a -> Id b -> Id c

(*>) :: Id a -> Id b -> Id b

(<*) :: Id a -> Id b -> Id a

Functor Id

Defined in Algebra.Types

Methods

fmap :: (a -> b) -> Id a -> Id b

(<$) :: a -> Id b -> Id a

Distributive Id

Defined in Algebra.Types

Methods

distribute :: Functor f => f (Id a) -> Id (f a)

collect :: Functor f => (a -> Id b) -> f a -> Id (f b)

distributeM :: Monad m => m (Id a) -> Id (m a)

collectM :: Monad m => (a -> Id b) -> m a -> Id (m b)

VectorR Id

Defined in Algebra.Linear

Methods

vectorSplit :: forall (f :: Type -> Type) (g :: Type -> Type). Id ~ (f ⊗ g) => Dict (VectorR f, VectorR g)

vectorCut :: forall (f :: Type -> Type) (g :: Type -> Type). Id ~ (f ⊕ g) => Dict (VectorR f, VectorR g)

Generic1 Id

Defined in Algebra.Types

Associated Types

type Rep1 Id :: k -> Type

Methods

from1 :: forall (a :: k). Id a -> Rep1 Id a

to1 :: forall (a :: k). Rep1 Id a -> Id a

Show x => Show (Id x)

Defined in Algebra.Types

Methods

showsPrec :: Int -> Id x -> ShowS

show :: Id x -> String

showList :: [Id x] -> ShowS

Eq x => Eq (Id x)

Defined in Algebra.Types

Methods

(==) :: Id x -> Id x -> Bool

(/=) :: Id x -> Id x -> Bool

Monoidal ((∘) :: (Type -> Type) -> (Type -> Type) -> Type -> Type) Id NatTrans

Defined in Algebra.Category.NatTrans

Methods

(⊗) :: forall (a :: k) (b :: k) (c :: k) (d :: k). (Obj NatTrans a, Obj NatTrans b, Obj NatTrans c, Obj NatTrans d) => NatTrans a b -> NatTrans c d -> NatTrans (a ∘ c) (b ∘ d)

assoc :: forall (a :: k) (b :: k) (c :: k). (Obj NatTrans a, Obj NatTrans b, Obj NatTrans c) => NatTrans ((a ∘ b) ∘ c) (a ∘ (b ∘ c))

assoc_ :: forall (a :: k) (b :: k) (c :: k). (Obj NatTrans a, Obj NatTrans b, Obj NatTrans c) => NatTrans (a ∘ (b ∘ c)) ((a ∘ b) ∘ c)

unitorR :: forall (a :: k). (Obj NatTrans a, Obj NatTrans Id) => NatTrans a (a ∘ Id)

unitorR_ :: forall (a :: k). (Obj NatTrans a, Obj NatTrans Id) => NatTrans (a ∘ Id) a

unitorL :: forall (a :: k). (Obj NatTrans a, Obj NatTrans Id) => NatTrans a (Id ∘ a)

unitorL_ :: forall (a :: k). (Obj NatTrans a, Obj NatTrans Id) => NatTrans (Id ∘ a) a

Ring s => Braided ((∘) :: (Type -> Type) -> (Type -> Type) -> Type -> Type) Id (Mat s :: (Type -> Type) -> (Type -> Type) -> Type)

Defined in Algebra.Linear

Methods

swap :: forall (a :: k) (b :: k). (Obj (Mat s) a, Obj (Mat s) b) => Mat s (a ∘ b) (b ∘ a)

swap_ :: forall (a :: k) (b :: k). (Obj (Mat s) a, Obj (Mat s) b) => Mat s (a ∘ b) (b ∘ a)

Ring s => Monoidal ((∘) :: (Type -> Type) -> (Type -> Type) -> Type -> Type) Id (Mat s :: (Type -> Type) -> (Type -> Type) -> Type)

Defined in Algebra.Linear

Methods

(⊗) :: forall (a :: k) (b :: k) (c :: k) (d :: k). (Obj (Mat s) a, Obj (Mat s) b, Obj (Mat s) c, Obj (Mat s) d) => Mat s a b -> Mat s c d -> Mat s (a ∘ c) (b ∘ d)

assoc :: forall (a :: k) (b :: k) (c :: k). (Obj (Mat s) a, Obj (Mat s) b, Obj (Mat s) c) => Mat s ((a ∘ b) ∘ c) (a ∘ (b ∘ c))

assoc_ :: forall (a :: k) (b :: k) (c :: k). (Obj (Mat s) a, Obj (Mat s) b, Obj (Mat s) c) => Mat s (a ∘ (b ∘ c)) ((a ∘ b) ∘ c)

unitorR :: forall (a :: k). (Obj (Mat s) a, Obj (Mat s) Id) => Mat s a (a ∘ Id)

unitorR_ :: forall (a :: k). (Obj (Mat s) a, Obj (Mat s) Id) => Mat s (a ∘ Id) a

unitorL :: forall (a :: k). (Obj (Mat s) a, Obj (Mat s) Id) => Mat s a (Id ∘ a)

unitorL_ :: forall (a :: k). (Obj (Mat s) a, Obj (Mat s) Id) => Mat s (Id ∘ a) a

Ring s => Symmetric ((∘) :: (Type -> Type) -> (Type -> Type) -> Type -> Type) Id (Mat s :: (Type -> Type) -> (Type -> Type) -> Type)

Defined in Algebra.Linear

type Rep Id

Defined in Algebra.Types

type Rep Id = One :: Type

type Rep1 Id

Defined in Algebra.Types

type Rep1 Id = D1 ('MetaData "Id" "Algebra.Types" "gasp-1.4.0.0-4Z31dYmTpJbCzuuYkBBWRm" 'True) (C1 ('MetaCons "Id" 'PrefixI 'True) (S1 ('MetaSel ('Just "fromId") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) Par1))

data CompClosed (con :: Type -> Constraint)

Constructors

CompClosed

Fields

• zero1Closed :: forall (x :: Type). Dict (con (One x))
• plus1Closed :: forall a b (x :: Type). (con (a x), con (b x)) => Dict (con ((a ⊗ b) x))
• one1Closed :: forall (x :: Type). con x => Dict (con (Id x))
• times1Closed :: forall (a :: Type -> Type) b (x :: Type). con (a (b x)) => Dict (con ((a ∘ b) x))

showCompClosed :: CompClosed Show