Safe Haskell	Safe-Inferred
Language	Haskell2010

Algebra.Types

Documentation

class SumKind k Source #

Associated Types

data (a :: k) ⊕ (b :: k) :: k Source #

data Zero :: k Source #

Instances

Instances details

SumKind Type Source #
Instance details Defined in Algebra.Types Associated Types data a ⊕ b :: k Source # data Zero :: k Source #
SumKind (Type -> Type) Source #
Instance details Defined in Algebra.Types Associated Types data a ⊕ b :: k Source # data Zero :: k Source #

class ProdKind k Source #

Associated Types

data (a :: k) ⊗ (b :: k) :: k Source #

data One :: k Source #

Instances

Instances details

ProdKind Type Source #
Instance details Defined in Algebra.Types Associated Types data a ⊗ b :: k Source # data One :: k Source #
ProdKind (Type -> Type) Source #
Instance details Defined in Algebra.Types Associated Types data a ⊗ b :: k Source # data One :: k Source #

class DualKind k Source #

Associated Types

data Dual (a :: k) :: k Source #

Instances

Instances details

DualKind Type Source #
Instance details Defined in Algebra.Types Associated Types data Dual a :: k Source #
DualKind (Type -> Type) Source #
Instance details Defined in Algebra.Types Associated Types data Dual a :: k Source #

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

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

Instances details

Arbitrary (Some1 (Repr x i t o)) Source #
Instance details 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) Source #
Instance details 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 Source #

type MRepr = Repr (⊗) One (⊕) Zero Source #

inhabitants :: Finite a => [a] Source #

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

Minimal complete definition

Nothing

Methods

typeSize :: Int Source #

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

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

Instances

Instances details

Finite Bool Source #
Instance details Defined in Algebra.Types Methods typeSize :: Int Source # finiteFstsnd :: Bool ~ (α ⊗ β) => Dict (Finite α, Finite β) Source # finiteLeftRight :: Bool ~ (α ⊕ β) => Dict (Finite α, Finite β) Source #
Finite (One :: Type) Source #
Instance details Defined in Algebra.Types Methods typeSize :: Int Source # finiteFstsnd :: One ~ (α ⊗ β) => Dict (Finite α, Finite β) Source # finiteLeftRight :: One ~ (α ⊕ β) => Dict (Finite α, Finite β) Source #
Finite (Zero :: Type) Source #
Instance details Defined in Algebra.Types Methods typeSize :: Int Source # finiteFstsnd :: Zero ~ (α ⊗ β) => Dict (Finite α, Finite β) Source # finiteLeftRight :: Zero ~ (α ⊕ β) => Dict (Finite α, Finite β) Source #
Finite a => Finite (Dual a) Source #
Instance details Defined in Algebra.Types Methods typeSize :: Int Source # finiteFstsnd :: Dual a ~ (α ⊗ β) => Dict (Finite α, Finite β) Source # finiteLeftRight :: Dual a ~ (α ⊕ β) => Dict (Finite α, Finite β) Source #
(Finite x, Finite y) => Finite (x ⊕ y) Source #
Instance details Defined in Algebra.Types Methods typeSize :: Int Source # finiteFstsnd :: (x ⊕ y) ~ (α ⊗ β) => Dict (Finite α, Finite β) Source # finiteLeftRight :: (x ⊕ y) ~ (α ⊕ β) => Dict (Finite α, Finite β) Source #
(Finite x, Finite y) => Finite (x ⊗ y) Source #
Instance details Defined in Algebra.Types Methods typeSize :: Int Source # finiteFstsnd :: (x ⊗ y) ~ (α ⊗ β) => Dict (Finite α, Finite β) Source # finiteLeftRight :: (x ⊗ y) ~ (α ⊕ β) => Dict (Finite α, Finite β) Source #

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

newtype (f ∘ g) x Source #

Constructors

Comp
Fields fromComp :: f (g x)

Instances

Instances details

Functor f => Generic1 (f ∘ g :: k -> Type) Source #
Instance details 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 Source #
Instance details 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) Source # assoc :: forall (a :: k) (b :: k) (c :: k). (Obj NatTrans a, Obj NatTrans b, Obj NatTrans c) => NatTrans ((a ∘ b) ∘ c) (a ∘ (b ∘ c)) Source # assoc_ :: forall (a :: k) (b :: k) (c :: k). (Obj NatTrans a, Obj NatTrans b, Obj NatTrans c) => NatTrans (a ∘ (b ∘ c)) ((a ∘ b) ∘ c) Source # unitorR :: forall (a :: k). (Obj NatTrans a, Obj NatTrans Id) => NatTrans a (a ∘ Id) Source # unitorR_ :: forall (a :: k). (Obj NatTrans a, Obj NatTrans Id) => NatTrans (a ∘ Id) a Source # unitorL :: forall (a :: k). (Obj NatTrans a, Obj NatTrans Id) => NatTrans a (Id ∘ a) Source # unitorL_ :: forall (a :: k). (Obj NatTrans a, Obj NatTrans Id) => NatTrans (Id ∘ a) a Source #
Ring s => Braided ((∘) :: (Type -> Type) -> (Type -> Type) -> Type -> Type) Id (Mat s :: (Type -> Type) -> (Type -> Type) -> Type) Source #
Instance details 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) Source # swap_ :: forall (a :: k) (b :: k). (Obj (Mat s) a, Obj (Mat s) b) => Mat s (a ∘ b) (b ∘ a) Source #
Ring s => Monoidal ((∘) :: (Type -> Type) -> (Type -> Type) -> Type -> Type) Id (Mat s :: (Type -> Type) -> (Type -> Type) -> Type) Source #
Instance details 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) Source # 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)) Source # 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) Source # unitorR :: forall (a :: k). (Obj (Mat s) a, Obj (Mat s) Id) => Mat s a (a ∘ Id) Source # unitorR_ :: forall (a :: k). (Obj (Mat s) a, Obj (Mat s) Id) => Mat s (a ∘ Id) a Source # unitorL :: forall (a :: k). (Obj (Mat s) a, Obj (Mat s) Id) => Mat s a (Id ∘ a) Source # unitorL_ :: forall (a :: k). (Obj (Mat s) a, Obj (Mat s) Id) => Mat s (Id ∘ a) a Source #
Ring s => Symmetric ((∘) :: (Type -> Type) -> (Type -> Type) -> Type -> Type) Id (Mat s :: (Type -> Type) -> (Type -> Type) -> Type) Source #
Instance details Defined in Algebra.Linear
(Arbitrary1 f, Arbitrary1 g) => Arbitrary1 (f ∘ g) Source #
Instance details 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) Source #
Instance details 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) Source #
Instance details 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) Source #
Instance details 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) Source #
Instance details 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) Source #
Instance details 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) Source #
Instance details 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) Source #
Instance details Defined in Algebra.Linear Methods vectorSplit :: forall (f :: Type -> Type) (g :: Type -> Type). (v ∘ w) ~ (f ⊗ g) => Dict (VectorR f, VectorR g) Source # vectorCut :: forall (f :: Type -> Type) (g :: Type -> Type). (v ∘ w) ~ (f ⊕ g) => Dict (VectorR f, VectorR g) Source #
Show (a (b x)) => Show ((a ∘ b) x) Source #
Instance details 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) Source #
Instance details 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) Source #
Instance details 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) Source #
Instance details Defined in Algebra.Types type Rep (v ∘ w) = Rep v ⊗ Rep w

newtype Id x Source #

Constructors

Id
Fields fromId :: x

Instances

Instances details

Arbitrary1 Id Source #
Instance details Defined in Algebra.Types Methods liftArbitrary :: Gen a -> Gen (Id a) # liftShrink :: (a -> [a]) -> Id a -> [Id a] #
Representable Id Source #
Instance details Defined in Algebra.Types Associated Types type Rep Id # Methods tabulate :: (Rep Id -> a) -> Id a # index :: Id a -> Rep Id -> a #
Foldable Id Source #
Instance details 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 Source #
Instance details 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 Source #
Instance details 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 Source #
Instance details Defined in Algebra.Types Methods fmap :: (a -> b) -> Id a -> Id b # (<$) :: a -> Id b -> Id a #
Distributive Id Source #
Instance details 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 Source #
Instance details Defined in Algebra.Linear Methods vectorSplit :: forall (f :: Type -> Type) (g :: Type -> Type). Id ~ (f ⊗ g) => Dict (VectorR f, VectorR g) Source # vectorCut :: forall (f :: Type -> Type) (g :: Type -> Type). Id ~ (f ⊕ g) => Dict (VectorR f, VectorR g) Source #
Generic1 Id Source #
Instance details 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) Source #
Instance details Defined in Algebra.Types Methods showsPrec :: Int -> Id x -> ShowS # show :: Id x -> String # showList :: [Id x] -> ShowS #
Eq x => Eq (Id x) Source #
Instance details Defined in Algebra.Types Methods (==) :: Id x -> Id x -> Bool # (/=) :: Id x -> Id x -> Bool #
Monoidal ((∘) :: (Type -> Type) -> (Type -> Type) -> Type -> Type) Id NatTrans Source #
Instance details 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) Source # assoc :: forall (a :: k) (b :: k) (c :: k). (Obj NatTrans a, Obj NatTrans b, Obj NatTrans c) => NatTrans ((a ∘ b) ∘ c) (a ∘ (b ∘ c)) Source # assoc_ :: forall (a :: k) (b :: k) (c :: k). (Obj NatTrans a, Obj NatTrans b, Obj NatTrans c) => NatTrans (a ∘ (b ∘ c)) ((a ∘ b) ∘ c) Source # unitorR :: forall (a :: k). (Obj NatTrans a, Obj NatTrans Id) => NatTrans a (a ∘ Id) Source # unitorR_ :: forall (a :: k). (Obj NatTrans a, Obj NatTrans Id) => NatTrans (a ∘ Id) a Source # unitorL :: forall (a :: k). (Obj NatTrans a, Obj NatTrans Id) => NatTrans a (Id ∘ a) Source # unitorL_ :: forall (a :: k). (Obj NatTrans a, Obj NatTrans Id) => NatTrans (Id ∘ a) a Source #
Ring s => Braided ((∘) :: (Type -> Type) -> (Type -> Type) -> Type -> Type) Id (Mat s :: (Type -> Type) -> (Type -> Type) -> Type) Source #
Instance details 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) Source # swap_ :: forall (a :: k) (b :: k). (Obj (Mat s) a, Obj (Mat s) b) => Mat s (a ∘ b) (b ∘ a) Source #
Ring s => Monoidal ((∘) :: (Type -> Type) -> (Type -> Type) -> Type -> Type) Id (Mat s :: (Type -> Type) -> (Type -> Type) -> Type) Source #
Instance details 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) Source # 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)) Source # 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) Source # unitorR :: forall (a :: k). (Obj (Mat s) a, Obj (Mat s) Id) => Mat s a (a ∘ Id) Source # unitorR_ :: forall (a :: k). (Obj (Mat s) a, Obj (Mat s) Id) => Mat s (a ∘ Id) a Source # unitorL :: forall (a :: k). (Obj (Mat s) a, Obj (Mat s) Id) => Mat s a (Id ∘ a) Source # unitorL_ :: forall (a :: k). (Obj (Mat s) a, Obj (Mat s) Id) => Mat s (Id ∘ a) a Source #
Ring s => Symmetric ((∘) :: (Type -> Type) -> (Type -> Type) -> Type -> Type) Id (Mat s :: (Type -> Type) -> (Type -> Type) -> Type) Source #
Instance details Defined in Algebra.Linear
type Rep Id Source #
Instance details Defined in Algebra.Types type Rep Id = One :: Type
type Rep1 Id Source #
Instance details 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) Source #

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 Source #