-{-# OPTIONS -XRankNTypes -XMultiParamTypeClasses -XNoMonomorphismRestriction -XTypeOperators -XFlexibleInstances -XFunctionalDependencies #-}
+{-# OPTIONS -XRankNTypes -XMultiParamTypeClasses -XNoMonomorphismRestriction -XTypeOperators -XFunctionalDependencies -XTypeFamilies -XFlexibleContexts #-}
-----------------------------------------------------------------------------
-- |
-- Module : GHC.HetMet.GArrow
GArrowConstant(..),
GArrowLiteral(..), -- should be implemented, but never invoked, by user code
- GArrowSum(..),
+ GArrowSum(..), ga_inl, ga_inr,
GArrowProd(..),
GArrowReify(..),
GArrowReflect(..),
GArrowCurry(..),
- GArrowApply(..)
+ GArrowApply(..),
+
+ GArrowTensor,
+ GArrowUnit,
+ GArrowExponent,
+
+ GArrowKappa(..),
+ GArrowSTKC(..),
+ GArrowSTKCL(..),
+ GArrowSTLC(..),
+ GArrowPCF(..)
+
) where
-import Control.Category
+import Control.Category hiding ((.))
+import Prelude hiding (id)
------------------------------------------------------------------------
-- The main GArrow class
-class Category g => GArrow g (**) u | (**) -> u where
+class Category g => GArrow g (**) u | (**) -> u, u -> (**) where
--id :: g x x
--comp :: g x y -> g y z -> g x z
ga_first :: g x y -> g (x ** z) (y ** z)
+
+
------------------------------------------------------------------------
-- Products, Coproducts, etc
-class (GArrow g (**) u,
- GArrow g (<*>) u) =>
- GArrowProd g (**) u (<*>) where
- ga_prod_copy :: g x (x<*>x)
- ga_prod_drop :: g x u
+class (GArrowDrop g (<*>) u,
+ GArrowCopy g (<*>) u) =>
+ GArrowProd g (<*>) u
-class (GArrow g (**) u,
- GArrow g (<+>) v) =>
- GArrowSum g (**) u v (<+>) where
- ga_merge :: g (x**x) x
- ga_never :: g v x
-
+class GArrow g (<+>) u =>
+ GArrowSum g (<+>) u where
+ ga_merge :: g (x<+>x) x
+ ga_never :: g u x
+ga_inl :: GArrowSum g (<+>) u => g x (x<+>y)
+ga_inl = ga_uncancelr >>> ga_second ga_never
+ga_inr :: GArrowSum g (<+>) u => g x (y<+>x)
+ga_inr = ga_uncancell >>> ga_first ga_never
------------------------------------------------------------------------
-- Loop
class GArrow g (**) u => GArrowLoop g (**) u where
- ga_loopl :: g (x**z) (y**z) -> g x y
- ga_loopr :: g (z**x) (z**y) -> g x y
+ ga_loopr :: g (x**z) (y**z) -> g x y
+ ga_loopl :: g (z**x) (z**y) -> g x y
------------------------------------------------------------------------
------------------------------------------------------------------------
--- Reify and Reflect, which are "curried" versions
+-- Reify and Reflect, which are "curried" versions of eval/const
-- If you have this for R the identity map on types, you're basically
-- a Control.Arrow; you can also define essentially all the other
------------------------------------------------------------------------
+-- The Kappa adjunction
+--
+-- See Hasegawa, Decomposing Typed Lambda Calculus into a Couple of
+-- Categorical Programming Languages) section 3, rule $(\times L)$
+
+class GArrow g (**) u => GArrowKappa g (**) u where
+ ga_kappa :: (g u x -> g u y) -> g x y
+
+
+
+
+
+------------------------------------------------------------------------
-- Apply and Curry
class GArrow g (**) u => GArrowApply g (**) u (~>) where
class GArrow g (**) u => GArrowCurry g (**) u (~>) where
ga_curryl :: g (x**y) z -> g x (y~>z)
ga_curryr :: g (x**y) z -> g y (x~>z)
+
+
+
+
+
+------------------------------------------------------------------------
+-- Type Families
+
+--
+-- The GArrow and GArrow{Copy,Drop,Swap} classes brandish their tensor
+-- and unit types; this is important because we might want to have
+-- both "instance GArrow g X Y" and "instance GArrow g Z Q" -- in
+-- fact, this is exactly how sums and pairs are defined.
+--
+-- However, in daily practice it's a pain to have all those extra type
+-- variables floating around. If you'd like to hide them, you can use
+-- the type families below to do so; see the definition of class
+-- GArrowSTKC for an example. Keep in mind, however, that any given
+-- type may only have a single instance declared using the type
+-- families.
+--
+
+type family GArrowTensor g :: * -> * -> * -- (**)
+type family GArrowUnit g :: * -- ()
+type family GArrowExponent g :: * -> * -> * -- (~>)
+
+
+
+
+------------------------------------------------------------------------
+-- Commonly Implemented Collections of Classes
+
+--
+-- The simply typed KAPPA calculus; see Hasegawa, __Decomposing Typed
+-- Lambda Calculus into a Couple of Categorical Programming
+-- Languages__, http://dx.doi.org/10.1007/3-540-60164-3_28
+--
+
+class (GArrowDrop g (GArrowTensor g) (GArrowUnit g),
+ GArrowCopy g (GArrowTensor g) (GArrowUnit g),
+ GArrowSwap g (GArrowTensor g) (GArrowUnit g)) =>
+ GArrowSTKC g
+
+class (GArrowDrop g (GArrowTensor g) (GArrowUnit g),
+ GArrowCopy g (GArrowTensor g) (GArrowUnit g),
+ GArrowSwap g (GArrowTensor g) (GArrowUnit g),
+ GArrowLoop g (GArrowTensor g) (GArrowUnit g)) =>
+ GArrowSTKCL g
+
+-- The simply typed LAMBDA calculus
+class (GArrowDrop g (GArrowTensor g) (GArrowUnit g),
+ GArrowCopy g (GArrowTensor g) (GArrowUnit g),
+ GArrowSwap g (GArrowTensor g) (GArrowUnit g),
+ GArrowCurry g (GArrowTensor g) (GArrowUnit g) (GArrowExponent g),
+ GArrowApply g (GArrowTensor g) (GArrowUnit g) (GArrowExponent g)
+ ) =>
+ GArrowSTLC g
+
+-- Programming Language for Computable Functions (w/o integers and booleans)
+class (GArrowDrop g (GArrowTensor g) (GArrowUnit g),
+ GArrowCopy g (GArrowTensor g) (GArrowUnit g),
+ GArrowSwap g (GArrowTensor g) (GArrowUnit g),
+ GArrowCurry g (GArrowTensor g) (GArrowUnit g) (GArrowExponent g),
+ GArrowApply g (GArrowTensor g) (GArrowUnit g) (GArrowExponent g),
+ GArrowLoop g (GArrowTensor g) (GArrowUnit g)
+ ) =>
+ GArrowPCF g (**) u (~>)
+
+
+
+
+
+------------------------------------------------------------------------
+-- Experimental, Not Yet Exported
+
+-- See Lindley, Wadler, and Yallop '08 -- except that here ga_force
+-- is primitive since there is no "arr" to define it in terms of.
+class GArrow g (**) u => GArrowStatic g (**) u (~>) where
+ ga_delay :: g a b -> g u (a~>b)
+ ga_force :: g u (a~>b) -> g a b
+ -- "ga_static/force_delay" forall a . force (delay a) = a
+ -- "ga_static/delay_force" forall a . delay (force a) = a
+
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