Agda-2.6.0.1: A dependently typed functional programming language and proof assistant

Safe Haskell | None
Language | Haskell2010

Agda.Utils.TypeLevel

Synopsis

- type family All (p :: k -> Constraint) (as :: [k]) :: Constraint where ...
- type family If (b :: Bool) (l :: k) (r :: k) :: k where ...
- type family Foldr (c :: k -> l -> l) (n :: l) (as :: [k]) :: l where ...
- type family Foldr' (c :: Function k (Function l l -> *) -> *) (n :: l) (as :: [k]) :: l where ...
- type family Map (f :: Function k l -> *) (as :: [k]) :: [l] where ...
- data ConsMap0 :: (Function k l -> *) -> Function k (Function [l] [l] -> *) -> *
- data ConsMap1 :: (Function k l -> *) -> k -> Function [l] [l] -> *
- type family Constant (b :: l) (as :: [k]) :: [l] where ...
- type Arrows (as :: [*]) (r :: *) = Foldr (->) r as
- type Products (as :: [*]) = Foldr (,) () as
- type family IsBase (t :: *) :: Bool where ...
- type family Domains (t :: *) :: [*] where ...
- type family Domains' (t :: *) :: [*] where ...
- type family CoDomain (t :: *) :: * where ...
- type family CoDomain' (t :: *) :: * where ...
- class Currying as b where
  - uncurrys :: Proxy as -> Proxy b -> Arrows as b -> Products as -> b
  - currys :: Proxy as -> Proxy b -> (Products as -> b) -> Arrows as b
- data Function :: * -> * -> *
- data Constant0 :: Function a (Function b a -> *) -> *
- data Constant1 :: * -> Function b a -> *
- type family Apply (t :: Function k l -> *) (u :: k) :: l

# Documentation

type family All (p :: k -> Constraint) (as :: [k]) :: Constraint where ... Source #

`All p as` ensures that the constraint `p` is satisfied by all the `types` in `as`. (Types is between scare-quotes here because the code is actually kind polymorphic)

Equations

All p '[] = () |
All p (a ': as) = (p a, All p as) |

type family If (b :: Bool) (l :: k) (r :: k) :: k where ... Source #

On Booleans

Equations

If True l r = l |
If False l r = r |

type family Foldr (c :: k -> l -> l) (n :: l) (as :: [k]) :: l where ... Source #

On Lists

Equations

Foldr c n '[] = n |
Foldr c n (a ': as) = c a (Foldr c n as) |

type family Foldr' (c :: Function k (Function l l -> *) -> *) (n :: l) (as :: [k]) :: l where ... Source #

Version of `Foldr` taking a defunctionalised argument so that we can use partially applied functions.

Equations

Foldr' c n '[] = n |
Foldr' c n (a ': as) = Apply (Apply c a) (Foldr' c n as) |

type family Map (f :: Function k l -> *) (as :: [k]) :: [l] where ... Source #

Equations

Map f as = Foldr' (ConsMap0 f) '[] as |

data ConsMap0 :: (Function k l -> *) -> Function k (Function [l] [l] -> *) -> * Source #

Instances

type Apply (ConsMap0 f :: Function k (Function [l] [l] -> Type) -> Type) (a :: k) Source # |
Instance details Defined in Agda.Utils.TypeLevel type Apply (ConsMap0 f :: Function k (Function [l] [l] -> Type) -> Type) (a :: k) = ConsMap1 f a

data ConsMap1 :: (Function k l -> *) -> k -> Function [l] [l] -> * Source #

Instances

type Apply (ConsMap1 f a2 :: Function [a1] [a1] -> Type) (tl :: [a1]) Source # |
Instance details Defined in Agda.Utils.TypeLevel type Apply (ConsMap1 f a2 :: Function [a1] [a1] -> Type) (tl :: [a1]) = Apply f a2 ': tl

type family Constant (b :: l) (as :: [k]) :: [l] where ... Source #

Equations

Constant b as = Map (Constant1 b) as |

type Arrows (as :: [*]) (r :: *) = Foldr (->) r as Source #

`Arrows [a1,..,an] r` corresponds to `a1 -> .. -> an -> r` | `Products [a1,..,an]` corresponds to `(a1, (..,( an, ())..))`

type Products (as :: [*]) = Foldr (,) () as Source #

type family IsBase (t :: *) :: Bool where ... Source #

`IsBase t` is `'True` whenever `t` is *not* a function space.

Equations

IsBase (a -> t) = False |
IsBase a = True |

type family Domains (t :: *) :: [*] where ... Source #

Using `IsBase` we can define notions of `Domains` and `CoDomains` which *reduce* under positive information `IsBase t ~ 'True` even though the shape of `t` is not formally exposed

Equations

Domains t = If (IsBase t) '[] (Domains' t) |

type family Domains' (t :: *) :: [*] where ... Source #

Equations

Domains' (a -> t) = a ': Domains t |

type family CoDomain (t :: *) :: * where ... Source #

Equations

CoDomain t = If (IsBase t) t (CoDomain' t) |

type family CoDomain' (t :: *) :: * where ... Source #

Equations

CoDomain' (a -> t) = CoDomain t |

class Currying as b where Source #

`Currying as b` witnesses the isomorphism between `Arrows as b` and `Products as -> b`. It is defined as a type class rather than by recursion on a singleton for `as` so all of that these conversions are inlined at compile time for concrete arguments.

Methods

uncurrys :: Proxy as -> Proxy b -> Arrows as b -> Products as -> b Source #

currys :: Proxy as -> Proxy b -> (Products as -> b) -> Arrows as b Source #

Instances

Currying ([] :: [Type]) b Source # |
Instance details Defined in Agda.Utils.TypeLevel Methods uncurrys :: Proxy [] -> Proxy b -> Arrows [] b -> Products [] -> b Source # currys :: Proxy [] -> Proxy b -> (Products [] -> b) -> Arrows [] b Source #
Currying as b => Currying (a ': as) b Source # |
Instance details Defined in Agda.Utils.TypeLevel Methods uncurrys :: Proxy (a ': as) -> Proxy b -> Arrows (a ': as) b -> Products (a ': as) -> b Source # currys :: Proxy (a ': as) -> Proxy b -> (Products (a ': as) -> b) -> Arrows (a ': as) b Source #

data Function :: * -> * -> * Source #

Instances

type Apply (ConsMap0 f :: Function k (Function [l] [l] -> Type) -> Type) (a :: k) Source # |
Instance details Defined in Agda.Utils.TypeLevel type Apply (ConsMap0 f :: Function k (Function [l] [l] -> Type) -> Type) (a :: k) = ConsMap1 f a
type Apply (Constant0 :: Function Type (Function b Type -> Type) -> Type) (a :: Type) Source # |
Instance details Defined in Agda.Utils.TypeLevel type Apply (Constant0 :: Function Type (Function b Type -> Type) -> Type) (a :: Type) = (Constant1 a :: Function b Type -> Type)

data Constant0 :: Function a (Function b a -> *) -> * Source #

Instances

type Apply (Constant0 :: Function Type (Function b Type -> Type) -> Type) (a :: Type) Source # |
Instance details Defined in Agda.Utils.TypeLevel type Apply (Constant0 :: Function Type (Function b Type -> Type) -> Type) (a :: Type) = (Constant1 a :: Function b Type -> Type)

data Constant1 :: * -> Function b a -> * Source #

Instances

type Apply (Constant1 a :: Function k Type -> Type) (b :: k) Source # |
Instance details Defined in Agda.Utils.TypeLevel type Apply (Constant1 a :: Function k Type -> Type) (b :: k) = a

type family Apply (t :: Function k l -> *) (u :: k) :: l Source #

Instances

type Apply (Constant1 a :: Function k Type -> Type) (b :: k) Source # |
Instance details Defined in Agda.Utils.TypeLevel type Apply (Constant1 a :: Function k Type -> Type) (b :: k) = a
type Apply (ConsMap1 f a2 :: Function [a1] [a1] -> Type) (tl :: [a1]) Source # |
Instance details Defined in Agda.Utils.TypeLevel type Apply (ConsMap1 f a2 :: Function [a1] [a1] -> Type) (tl :: [a1]) = Apply f a2 ': tl
type Apply (ConsMap0 f :: Function k (Function [l] [l] -> Type) -> Type) (a :: k) Source # |
Instance details Defined in Agda.Utils.TypeLevel type Apply (ConsMap0 f :: Function k (Function [l] [l] -> Type) -> Type) (a :: k) = ConsMap1 f a
type Apply (Constant0 :: Function Type (Function b Type -> Type) -> Type) (a :: Type) Source # |
Instance details Defined in Agda.Utils.TypeLevel type Apply (Constant0 :: Function Type (Function b Type -> Type) -> Type) (a :: Type) = (Constant1 a :: Function b Type -> Type)