{-# LANGUAGE FunctionalDependencies #-}
{-# LANGUAGE UndecidableInstances #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE StandaloneDeriving #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE ConstraintKinds #-}
{-# LANGUAGE EmptyDataDecls #-}
{-# LANGUAGE KindSignatures #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE Trustworthy #-}
{-# LANGUAGE Rank2Types #-}
{-# LANGUAGE GADTs #-}
{-# LANGUAGE CPP #-}
#if __GLASGOW_HASKELL__ >= 707
{-# LANGUAGE AllowAmbiguousTypes #-}
{-# LANGUAGE DeriveDataTypeable #-}
{-# LANGUAGE RoleAnnotations #-}
#endif
-----------------------------------------------------------------------------
-- |
-- Module      :  Data.Constraint
-- Copyright   :  (C) 2011-2015 Edward Kmett,
-- License     :  BSD-style (see the file LICENSE)
--
-- Maintainer  :  Edward Kmett <ekmett@gmail.com>
-- Stability   :  experimental
-- Portability :  non-portable
--
-- @ConstraintKinds@ made type classes into types of a new kind, @Constraint@.
--
-- @
-- 'Eq' :: * -> 'Constraint'
-- 'Ord' :: * -> 'Constraint'
-- 'Monad' :: (* -> *) -> 'Constraint'
-- @
--
-- The need for this extension was first publicized in the paper
--
-- <http://research.microsoft.com/pubs/67439/gmap3.pdf Scrap your boilerplate with class: extensible generic functions>
--
-- by Ralf Lämmel and Simon Peyton Jones in 2005, which shoehorned all the
-- things they needed into a custom 'Sat' typeclass.
--
-- With @ConstraintKinds@ we can put into code a lot of tools for manipulating
-- these new types without such awkward workarounds.
----------------------------------------------------------------------------
module Data.Constraint
  (
  -- * The Kind of Constraints
    Constraint
  -- * Dictionary
  , Dict(Dict)
  -- * Entailment
  , (:-)(Sub)
  , (\\)
  , weaken1, weaken2, contract
  , (&&&), (***)
  , trans, refl
  , top, bottom
  -- * Dict is fully faithful
  , mapDict
  , unmapDict
  -- * Reflection
  , Class(..)
  , (:=>)(..)
  ) where
import Control.Monad
#if __GLASGOW_HASKELL__ >= 707
import Control.Category
#endif
import Control.Applicative
#if __GLASGOW_HASKELL__ < 710
import Data.Monoid
#endif
import Data.Complex
import Data.Ratio
#if __GLASGOW_HASKELL__ >= 707
import Data.Data
#endif
import GHC.Prim (Constraint)

-- | Values of type @'Dict' p@ capture a dictionary for a constraint of type @p@.
--
-- e.g.
--
-- @
-- 'Dict' :: 'Dict' ('Eq' 'Int')
-- @
--
-- captures a dictionary that proves we have an:
--
-- @
-- instance 'Eq' 'Int
-- @
--
-- Pattern matching on the 'Dict' constructor will bring this instance into scope.
--
data Dict :: Constraint -> * where
  Dict :: a => Dict a
#if __GLASGOW_HASKELL__ >= 707
  deriving Typeable


instance (Typeable p, p) => Data (Dict p) where
  gfoldl _ z Dict = z Dict
  toConstr _ = dictConstr
  gunfold _ z c = case constrIndex c of
    1 -> z Dict
    _ -> error "gunfold"
  dataTypeOf _ = dictDataType

dictConstr :: Constr
dictConstr = mkConstr dictDataType "Dict" [] Prefix

dictDataType :: DataType
dictDataType = mkDataType "Data.Constraint.Dict" [dictConstr]
#endif

deriving instance Eq (Dict a)
deriving instance Ord (Dict a)
deriving instance Show (Dict a)

infixr 9 :-

-- | This is the type of entailment.
--
-- @a ':-' b@ is read as @a@ \"entails\" @b@.
--
-- With this we can actually build a category for 'Constraint' resolution.
--
-- e.g.
--
-- Because @'Eq' a@ is a superclass of @'Ord' a@, we can show that @'Ord' a@
-- entails @'Eq' a@.
--
-- Because @instance 'Ord' a => 'Ord' [a]@ exists, we can show that @'Ord' a@
-- entails @'Ord' [a]@ as well.
--
-- This relationship is captured in the ':-' entailment type here.
--
-- Since @p ':-' p@ and entailment composes, ':-' forms the arrows of a 'Category'
-- of constraints. However, 'Category' only because sufficiently general to support this
-- instance in GHC 7.8, so prior to 7.8 this instance is unavailable.
--
-- But due to the coherence of instance resolution in Haskell, this 'Category'
-- has some very interesting properties. Notably, in the absence of
-- @IncoherentInstances@, this category is \"thin\", which is to say that
-- between any two objects (constraints) there is at most one distinguishable
-- arrow.
--
-- This means that for instance, even though there are two ways to derive
-- @'Ord' a ':-' 'Eq' [a]@, the answers from these two paths _must_ by
-- construction be equal. This is a property that Haskell offers that is
-- pretty much unique in the space of languages with things they call \"type
-- classes\".
--
-- What are the two ways?
--
-- Well, we can go from @'Ord' a ':-' 'Eq' a@ via the
-- superclass relationship, and them from @'Eq' a ':-' 'Eq' [a]@ via the
-- instance, or we can go from @'Ord' a ':-' 'Ord' [a]@ via the instance
-- then from @'Ord' [a] ':-' 'Eq' [a]@ through the superclass relationship
-- and this diagram by definition must \"commute\".
--
-- Diagrammatically,
--
-- >                    Ord a
-- >                ins /     \ cls
-- >                   v       v
-- >             Ord [a]     Eq a
-- >                cls \     / ins
-- >                     v   v
-- >                    Eq [a]
--
-- This safety net ensures that pretty much anything you can write with this
-- library is sensible and can't break any assumptions on the behalf of
-- library authors.
newtype a :- b = Sub (a => Dict b)
#if __GLASGOW_HASKELL__ >= 707
  deriving Typeable

type role (:-) nominal nominal

-- TODO: _proper_ Data for @(p ':-' q)@ requires @(:-)@ to be cartesian _closed_.
--
-- This is admissable, but not present by default

-- constraint should be instance (Typeable p, Typeable q, p |- q) => Data (p :- q)
instance (Typeable p, Typeable q, p, q) => Data (p :- q) where
  gfoldl _ z (Sub Dict) = z (Sub Dict)
  toConstr _ = subConstr
  gunfold _ z c = case constrIndex c of
    1 -> z (Sub Dict)
    _ -> error "gunfold"
  dataTypeOf _ = subDataType

subConstr :: Constr
subConstr = mkConstr dictDataType "Sub" [] Prefix

subDataType :: DataType
subDataType = mkDataType "Data.Constraint.:-" [subConstr]

-- | Possible since GHC 7.8, when 'Category' was made polykinded.
instance Category (:-) where
  id  = refl
  (.) = trans
#endif

-- | Assumes 'IncoherentInstances' doesn't exist.
instance Eq (a :- b) where
  _ == _ = True

-- | Assumes 'IncoherentInstances' doesn't exist.
instance Ord (a :- b) where
  compare _ _ = EQ

instance Show (a :- b) where
  showsPrec d _ = showParen (d > 10) $ showString "Sub Dict"

infixl 1 \\ -- required comment

-- | Given that @a :- b@, derive something that needs a context @b@, using the context @a@
(\\) :: a => (b => r) -> (a :- b) -> r
r \\ Sub Dict = r

--------------------------------------------------------------------------------
-- Constraints form a Category
--------------------------------------------------------------------------------

-- | Transitivity of entailment
--
-- If we view @(':-')@ as a Constraint-indexed category, then this is @('.')@
trans :: (b :- c) -> (a :- b) -> a :- c
trans f g = Sub $ Dict \\ f \\ g

-- | Reflexivity of entailment
--
-- If we view @(':-')@ as a Constraint-indexed category, then this is 'id'
refl :: a :- a
refl = Sub Dict

--------------------------------------------------------------------------------
-- (,) is a Bifunctor
--------------------------------------------------------------------------------

-- | due to the hack for the kind of @(,)@ in the current version of GHC we can't actually
-- make instances for @(,) :: Constraint -> Constraint -> Constraint@, but @(,)@ is a
-- bifunctor on the category of constraints. This lets us map over both sides.
(***) :: (a :- b) -> (c :- d) -> (a, c) :- (b, d)
f *** g = Sub $ Dict \\ f \\ g

--------------------------------------------------------------------------------
-- Constraints are Cartesian
--------------------------------------------------------------------------------

-- | Weakening a constraint product
--
-- The category of constraints is Cartesian. We can forget information.
weaken1 :: (a, b) :- a
weaken1 = Sub Dict

-- | Weakening a constraint product
--
-- The category of constraints is Cartesian. We can forget information.
weaken2 :: (a, b) :- b
weaken2 = Sub Dict

-- | Contracting a constraint / diagonal morphism
--
-- The category of constraints is Cartesian. We can reuse information.
contract :: a :- (a, a)
contract = Sub Dict

-- | Constraint product
--
-- > trans weaken1 (f &&& g) = f
-- > trans weaken2 (f &&& g) = g
(&&&) :: (a :- b) -> (a :- c) -> a :- (b, c)
f &&& g = Sub $ Dict \\ f \\ g

--------------------------------------------------------------------------------
-- Initial and terminal morphisms
--------------------------------------------------------------------------------

-- | Every constraint implies truth
--
-- These are the terminal arrows of the category, and @()@ is the terminal object.
--
-- Given any constraint there is a unique entailment of the @()@ constraint from that constraint.
top :: a :- ()
top = Sub Dict

type family Ex (a :: *) (c :: Constraint) :: Constraint
type instance Ex () c = ()
type instance Ex Bool c = c

falso :: (() ~ a) :- Ex a c
falso = Sub Dict

-- |
-- A bad type coercion lets you derive any constraint you want.
--
-- These are the initial arrows of the category and @(() ~ Bool)@ is the initial object
--
-- This demonstrates the law of classical logic <http://en.wikipedia.org/wiki/Principle_of_explosion "ex falso quodlibet">
bottom :: (() ~ Bool) :- c
bottom = falso

--------------------------------------------------------------------------------
-- Dict is fully faithful
--------------------------------------------------------------------------------

-- | Apply an entailment to a dictionary.
--
-- From a category theoretic perspective 'Dict' is a functor that maps from the category
-- of constraints (with arrows in ':-') to the category Hask of Haskell data types.
mapDict :: (a :- b) -> Dict a -> Dict b
mapDict p Dict = case p of Sub q -> q

-- |
-- This functor is fully faithful, which is to say that given any function you can write
-- @Dict a -> Dict b@ there also exists an entailment @a :- b@ in the category of constraints
-- that you can build.
unmapDict :: (Dict a -> Dict b) -> a :- b
unmapDict f = Sub (f Dict)

#if __GLASGOW_HASKELL__ >= 707
type role Dict nominal
#endif

--------------------------------------------------------------------------------
-- Reflection
--------------------------------------------------------------------------------

-- | Reify the relationship between a class and its superclass constraints as a class
--
-- Given a definition such as
--
-- @
-- class Foo a => Bar a
-- @
--
-- you can capture the relationship between 'Bar a' and its superclass 'Foo a' with
--
-- @
-- instance 'Class' (Foo a) (Bar a) where 'cls' = 'Sub' 'Dict'
-- @
--
-- Now the user can use 'cls :: Bar a :- Foo a'
class Class b h | h -> b where
  cls :: h :- b

infixr 9 :=>
-- | Reify the relationship between an instance head and its body as a class
--
-- Given a definition such as
--
-- @
-- instance Foo a => Foo [a]
-- @
--
-- you can capture the relationship between the instance head and its body with
--
-- @
-- instance Foo a ':=>' Foo [a] where 'ins' = 'Sub' 'Dict'
-- @
class b :=> h | h -> b where
  ins :: b :- h

-- Bootstrapping

instance Class () (Class b a) where cls = Sub Dict
instance Class () (b :=> a) where cls = Sub Dict

instance Class b a => () :=> Class b a where ins = Sub Dict
instance (b :=> a) => () :=> b :=> a where ins = Sub Dict

instance Class () () where cls = Sub Dict
instance () :=> () where ins = Sub Dict

-- Local, Prelude, Applicative, C.M.I and Data.Monoid instances

-- Eq
instance Class () (Eq a) where cls = Sub Dict
instance () :=> Eq () where ins = Sub Dict
instance () :=> Eq Int where ins = Sub Dict
instance () :=> Eq Bool where ins = Sub Dict
instance () :=> Eq Integer where ins = Sub Dict
instance () :=> Eq Float where ins = Sub Dict
instance () :=> Eq Double where ins = Sub Dict
instance Eq a :=> Eq [a] where ins = Sub Dict
instance Eq a :=> Eq (Maybe a) where ins = Sub Dict
instance Eq a :=> Eq (Complex a) where ins = Sub Dict
instance Eq a :=> Eq (Ratio a) where ins = Sub Dict
instance (Eq a, Eq b) :=> Eq (a, b) where ins = Sub Dict
instance (Eq a, Eq b) :=> Eq (Either a b) where ins = Sub Dict
instance () :=> Eq (Dict a) where ins = Sub Dict
instance () :=> Eq (a :- b) where ins = Sub Dict

-- Ord
instance Class (Eq a) (Ord a) where cls = Sub Dict
instance () :=> Ord () where ins = Sub Dict
instance () :=> Ord Bool where ins = Sub Dict
instance () :=> Ord Int where ins = Sub Dict
instance ():=> Ord Integer where ins = Sub Dict
instance () :=> Ord Float where ins = Sub Dict
instance ():=> Ord Double where ins = Sub Dict
instance () :=> Ord Char where ins = Sub Dict
instance Ord a :=> Ord (Maybe a) where ins = Sub Dict
instance Ord a :=> Ord [a] where ins = Sub Dict
instance (Ord a, Ord b) :=> Ord (a, b) where ins = Sub Dict
instance (Ord a, Ord b) :=> Ord (Either a b) where ins = Sub Dict
instance Integral a :=> Ord (Ratio a) where ins = Sub Dict
instance () :=> Ord (Dict a) where ins = Sub Dict
instance () :=> Ord (a :- b) where ins = Sub Dict

-- Show
instance Class () (Show a) where cls = Sub Dict
instance () :=> Show () where ins = Sub Dict
instance () :=> Show Bool where ins = Sub Dict
instance () :=> Show Ordering where ins = Sub Dict
instance () :=> Show Char where ins = Sub Dict
instance Show a :=> Show (Complex a) where ins = Sub Dict
instance Show a :=> Show [a] where ins = Sub Dict
instance Show a :=> Show (Maybe a) where ins = Sub Dict
instance (Show a, Show b) :=> Show (a, b) where ins = Sub Dict
instance (Show a, Show b) :=> Show (Either a b) where ins = Sub Dict
instance (Integral a, Show a) :=> Show (Ratio a) where ins = Sub Dict
instance () :=> Show (Dict a) where ins = Sub Dict
instance () :=> Show (a :- b) where ins = Sub Dict

-- Read
instance Class () (Read a) where cls = Sub Dict
instance () :=> Read () where ins = Sub Dict
instance () :=> Read Bool where ins = Sub Dict
instance () :=> Read Ordering where ins = Sub Dict
instance () :=> Read Char where ins = Sub Dict
instance Read a :=> Read (Complex a) where ins = Sub Dict
instance Read a :=> Read [a] where ins = Sub Dict
instance Read a :=> Read (Maybe a) where ins = Sub Dict
instance (Read a, Read b) :=> Read (a, b) where ins = Sub Dict
instance (Read a, Read b) :=> Read (Either a b) where ins = Sub Dict
instance (Integral a, Read a) :=> Read (Ratio a) where ins = Sub Dict

-- Enum
instance Class () (Enum a) where cls = Sub Dict
instance () :=> Enum () where ins = Sub Dict
instance () :=> Enum Bool where ins = Sub Dict
instance () :=> Enum Ordering where ins = Sub Dict
instance () :=> Enum Char where ins = Sub Dict
instance () :=> Enum Int where ins = Sub Dict
instance () :=> Enum Integer where ins = Sub Dict
instance () :=> Enum Float where ins = Sub Dict
instance () :=> Enum Double where ins = Sub Dict
instance Integral a :=> Enum (Ratio a) where ins = Sub Dict

-- Bounded
instance Class () (Bounded a) where cls = Sub Dict
instance () :=> Bounded () where ins = Sub Dict
instance () :=> Bounded Ordering where ins = Sub Dict
instance () :=> Bounded Bool where ins = Sub Dict
instance () :=> Bounded Int where ins = Sub Dict
instance () :=> Bounded Char where ins = Sub Dict
instance (Bounded a, Bounded b) :=> Bounded (a,b) where ins = Sub Dict

-- Num
instance Class () (Num a) where cls = Sub Dict
instance () :=> Num Int where ins = Sub Dict
instance () :=> Num Integer where ins = Sub Dict
instance () :=> Num Float where ins = Sub Dict
instance () :=> Num Double where ins = Sub Dict
instance RealFloat a :=> Num (Complex a) where ins = Sub Dict
instance Integral a :=> Num (Ratio a) where ins = Sub Dict

-- Real
instance Class (Num a, Ord a) (Real a) where cls = Sub Dict
instance () :=> Real Int where ins = Sub Dict
instance () :=> Real Integer where ins = Sub Dict
instance () :=> Real Float where ins = Sub Dict
instance () :=> Real Double where ins = Sub Dict
instance Integral a :=> Real (Ratio a) where ins = Sub Dict

-- Integral
instance Class (Real a, Enum a) (Integral a) where cls = Sub Dict
instance () :=> Integral Int where ins = Sub Dict
instance () :=> Integral Integer where ins = Sub Dict

-- Fractional
instance Class (Num a) (Fractional a) where cls = Sub Dict
instance () :=> Fractional Float where ins = Sub Dict
instance () :=> Fractional Double where ins = Sub Dict
instance RealFloat a :=> Fractional (Complex a) where ins = Sub Dict
instance Integral a :=> Fractional (Ratio a) where ins = Sub Dict

-- Floating
instance Class (Fractional a) (Floating a) where cls = Sub Dict
instance () :=> Floating Float where ins = Sub Dict
instance () :=> Floating Double where ins = Sub Dict
instance RealFloat a :=> Floating (Complex a) where ins = Sub Dict

-- RealFrac
instance Class (Real a, Fractional a) (RealFrac a) where cls = Sub Dict
instance () :=> RealFrac Float where ins = Sub Dict
instance () :=> RealFrac Double where ins = Sub Dict
instance Integral a :=> RealFrac (Ratio a) where ins = Sub Dict

-- RealFloat
instance Class (RealFrac a, Floating a) (RealFloat a) where cls = Sub Dict
instance () :=> RealFloat Float where ins = Sub Dict
instance () :=> RealFloat Double where ins = Sub Dict

-- Monoid
instance Class () (Monoid a) where cls = Sub Dict
instance () :=> Monoid () where ins = Sub Dict
instance () :=> Monoid Ordering where ins = Sub Dict
instance () :=> Monoid [a] where ins = Sub Dict
instance Monoid a :=> Monoid (Maybe a) where ins = Sub Dict
instance (Monoid a, Monoid b) :=> Monoid (a, b) where ins = Sub Dict

-- Functor
instance Class () (Functor f) where cls = Sub Dict
instance () :=> Functor [] where ins = Sub Dict
instance () :=> Functor Maybe where ins = Sub Dict
instance () :=> Functor (Either a) where ins = Sub Dict
instance () :=> Functor ((->) a) where ins = Sub Dict
instance () :=> Functor ((,) a) where ins = Sub Dict
instance () :=> Functor IO where ins = Sub Dict
instance Monad m :=> Functor (WrappedMonad m) where ins = Sub Dict

-- Applicative
instance Class (Functor f) (Applicative f) where cls = Sub Dict
instance () :=> Applicative [] where ins = Sub Dict
instance () :=> Applicative Maybe where ins = Sub Dict
instance () :=> Applicative (Either a) where ins = Sub Dict
instance () :=> Applicative ((->)a) where ins = Sub Dict
instance () :=> Applicative IO where ins = Sub Dict
instance Monoid a :=> Applicative ((,)a) where ins = Sub Dict
instance Monad m :=> Applicative (WrappedMonad m) where ins = Sub Dict

-- Alternative
instance Class (Applicative f) (Alternative f) where cls = Sub Dict
instance () :=> Alternative [] where ins = Sub Dict
instance () :=> Alternative Maybe where ins = Sub Dict
instance MonadPlus m :=> Alternative (WrappedMonad m) where ins = Sub Dict

-- Monad
instance Class () (Monad f) where cls = Sub Dict
instance () :=> Monad [] where ins = Sub Dict
instance () :=> Monad ((->) a) where ins = Sub Dict
instance () :=> Monad (Either a) where ins = Sub Dict
instance () :=> Monad IO where ins = Sub Dict

-- MonadPlus
instance Class (Monad f) (MonadPlus f) where cls = Sub Dict
instance () :=> MonadPlus [] where ins = Sub Dict
instance () :=> MonadPlus Maybe where ins = Sub Dict

--------------------------------------------------------------------------------
-- UndecidableInstances
--------------------------------------------------------------------------------

instance a :=> Enum (Dict a) where ins = Sub Dict
instance a => Enum (Dict a) where
  toEnum _ = Dict
  fromEnum Dict = 0

instance a :=> Bounded (Dict a) where ins = Sub Dict
instance a => Bounded (Dict a) where
  minBound = Dict
  maxBound = Dict

instance a :=> Read (Dict a) where ins = Sub Dict
deriving instance a => Read (Dict a)

instance a :=> Monoid (Dict a) where ins = Sub Dict
instance a => Monoid (Dict a) where
  mappend Dict Dict = Dict
  mempty = Dict