{-# LANGUAGE DeriveDataTypeable #-}
{-# LANGUAGE DeriveFoldable #-}
{-# LANGUAGE DeriveFunctor #-}
{-# LANGUAGE DeriveGeneric #-}
{-# LANGUAGE DeriveTraversable #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE Safe #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE TypeOperators #-}
module Algebra.Lattice.Ordered (
Ordered(..)
) where
import Prelude ()
import Prelude.Compat
import Algebra.Heyting
import Algebra.Lattice
import Algebra.PartialOrd
import Control.DeepSeq (NFData (..))
import Control.Monad (ap)
import Data.Data (Data, Typeable)
import Data.Hashable (Hashable (..))
import Data.Universe.Class (Finite (..), Universe (..))
import Data.Universe.Helpers (Natural, Tagged, retag)
import GHC.Generics (Generic, Generic1)
import qualified Test.QuickCheck as QC
newtype Ordered a = Ordered { forall a. Ordered a -> a
getOrdered :: a }
deriving ( Ordered a -> Ordered a -> Bool
forall a. Eq a => Ordered a -> Ordered a -> Bool
forall a. (a -> a -> Bool) -> (a -> a -> Bool) -> Eq a
/= :: Ordered a -> Ordered a -> Bool
$c/= :: forall a. Eq a => Ordered a -> Ordered a -> Bool
== :: Ordered a -> Ordered a -> Bool
$c== :: forall a. Eq a => Ordered a -> Ordered a -> Bool
Eq, Ordered a -> Ordered a -> Bool
Ordered a -> Ordered a -> Ordering
Ordered a -> Ordered a -> Ordered a
forall a.
Eq a
-> (a -> a -> Ordering)
-> (a -> a -> Bool)
-> (a -> a -> Bool)
-> (a -> a -> Bool)
-> (a -> a -> Bool)
-> (a -> a -> a)
-> (a -> a -> a)
-> Ord a
forall {a}. Ord a => Eq (Ordered a)
forall a. Ord a => Ordered a -> Ordered a -> Bool
forall a. Ord a => Ordered a -> Ordered a -> Ordering
forall a. Ord a => Ordered a -> Ordered a -> Ordered a
min :: Ordered a -> Ordered a -> Ordered a
$cmin :: forall a. Ord a => Ordered a -> Ordered a -> Ordered a
max :: Ordered a -> Ordered a -> Ordered a
$cmax :: forall a. Ord a => Ordered a -> Ordered a -> Ordered a
>= :: Ordered a -> Ordered a -> Bool
$c>= :: forall a. Ord a => Ordered a -> Ordered a -> Bool
> :: Ordered a -> Ordered a -> Bool
$c> :: forall a. Ord a => Ordered a -> Ordered a -> Bool
<= :: Ordered a -> Ordered a -> Bool
$c<= :: forall a. Ord a => Ordered a -> Ordered a -> Bool
< :: Ordered a -> Ordered a -> Bool
$c< :: forall a. Ord a => Ordered a -> Ordered a -> Bool
compare :: Ordered a -> Ordered a -> Ordering
$ccompare :: forall a. Ord a => Ordered a -> Ordered a -> Ordering
Ord, Int -> Ordered a -> ShowS
forall a. Show a => Int -> Ordered a -> ShowS
forall a. Show a => [Ordered a] -> ShowS
forall a. Show a => Ordered a -> String
forall a.
(Int -> a -> ShowS) -> (a -> String) -> ([a] -> ShowS) -> Show a
showList :: [Ordered a] -> ShowS
$cshowList :: forall a. Show a => [Ordered a] -> ShowS
show :: Ordered a -> String
$cshow :: forall a. Show a => Ordered a -> String
showsPrec :: Int -> Ordered a -> ShowS
$cshowsPrec :: forall a. Show a => Int -> Ordered a -> ShowS
Show, ReadPrec [Ordered a]
ReadPrec (Ordered a)
ReadS [Ordered a]
forall a. Read a => ReadPrec [Ordered a]
forall a. Read a => ReadPrec (Ordered a)
forall a. Read a => Int -> ReadS (Ordered a)
forall a. Read a => ReadS [Ordered a]
forall a.
(Int -> ReadS a)
-> ReadS [a] -> ReadPrec a -> ReadPrec [a] -> Read a
readListPrec :: ReadPrec [Ordered a]
$creadListPrec :: forall a. Read a => ReadPrec [Ordered a]
readPrec :: ReadPrec (Ordered a)
$creadPrec :: forall a. Read a => ReadPrec (Ordered a)
readList :: ReadS [Ordered a]
$creadList :: forall a. Read a => ReadS [Ordered a]
readsPrec :: Int -> ReadS (Ordered a)
$creadsPrec :: forall a. Read a => Int -> ReadS (Ordered a)
Read, Ordered a -> DataType
Ordered a -> Constr
forall {a}. Data a => Typeable (Ordered a)
forall a. Data a => Ordered a -> DataType
forall a. Data a => Ordered a -> Constr
forall a.
Data a =>
(forall b. Data b => b -> b) -> Ordered a -> Ordered a
forall a u.
Data a =>
Int -> (forall d. Data d => d -> u) -> Ordered a -> u
forall a u.
Data a =>
(forall d. Data d => d -> u) -> Ordered a -> [u]
forall a r r'.
Data a =>
(r -> r' -> r)
-> r -> (forall d. Data d => d -> r') -> Ordered a -> r
forall a r r'.
Data a =>
(r' -> r -> r)
-> r -> (forall d. Data d => d -> r') -> Ordered a -> r
forall a (m :: * -> *).
(Data a, Monad m) =>
(forall d. Data d => d -> m d) -> Ordered a -> m (Ordered a)
forall a (m :: * -> *).
(Data a, MonadPlus m) =>
(forall d. Data d => d -> m d) -> Ordered a -> m (Ordered a)
forall a (c :: * -> *).
Data a =>
(forall b r. Data b => c (b -> r) -> c r)
-> (forall r. r -> c r) -> Constr -> c (Ordered a)
forall a (c :: * -> *).
Data a =>
(forall d b. Data d => c (d -> b) -> d -> c b)
-> (forall g. g -> c g) -> Ordered a -> c (Ordered a)
forall a (t :: * -> *) (c :: * -> *).
(Data a, Typeable t) =>
(forall d. Data d => c (t d)) -> Maybe (c (Ordered a))
forall a (t :: * -> * -> *) (c :: * -> *).
(Data a, Typeable t) =>
(forall d e. (Data d, Data e) => c (t d e))
-> Maybe (c (Ordered a))
forall a.
Typeable a
-> (forall (c :: * -> *).
(forall d b. Data d => c (d -> b) -> d -> c b)
-> (forall g. g -> c g) -> a -> c a)
-> (forall (c :: * -> *).
(forall b r. Data b => c (b -> r) -> c r)
-> (forall r. r -> c r) -> Constr -> c a)
-> (a -> Constr)
-> (a -> DataType)
-> (forall (t :: * -> *) (c :: * -> *).
Typeable t =>
(forall d. Data d => c (t d)) -> Maybe (c a))
-> (forall (t :: * -> * -> *) (c :: * -> *).
Typeable t =>
(forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c a))
-> ((forall b. Data b => b -> b) -> a -> a)
-> (forall r r'.
(r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> a -> r)
-> (forall r r'.
(r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> a -> r)
-> (forall u. (forall d. Data d => d -> u) -> a -> [u])
-> (forall u. Int -> (forall d. Data d => d -> u) -> a -> u)
-> (forall (m :: * -> *).
Monad m =>
(forall d. Data d => d -> m d) -> a -> m a)
-> (forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> a -> m a)
-> (forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> a -> m a)
-> Data a
forall (c :: * -> *).
(forall b r. Data b => c (b -> r) -> c r)
-> (forall r. r -> c r) -> Constr -> c (Ordered a)
forall (c :: * -> *).
(forall d b. Data d => c (d -> b) -> d -> c b)
-> (forall g. g -> c g) -> Ordered a -> c (Ordered a)
forall (t :: * -> *) (c :: * -> *).
Typeable t =>
(forall d. Data d => c (t d)) -> Maybe (c (Ordered a))
gmapMo :: forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> Ordered a -> m (Ordered a)
$cgmapMo :: forall a (m :: * -> *).
(Data a, MonadPlus m) =>
(forall d. Data d => d -> m d) -> Ordered a -> m (Ordered a)
gmapMp :: forall (m :: * -> *).
MonadPlus m =>
(forall d. Data d => d -> m d) -> Ordered a -> m (Ordered a)
$cgmapMp :: forall a (m :: * -> *).
(Data a, MonadPlus m) =>
(forall d. Data d => d -> m d) -> Ordered a -> m (Ordered a)
gmapM :: forall (m :: * -> *).
Monad m =>
(forall d. Data d => d -> m d) -> Ordered a -> m (Ordered a)
$cgmapM :: forall a (m :: * -> *).
(Data a, Monad m) =>
(forall d. Data d => d -> m d) -> Ordered a -> m (Ordered a)
gmapQi :: forall u. Int -> (forall d. Data d => d -> u) -> Ordered a -> u
$cgmapQi :: forall a u.
Data a =>
Int -> (forall d. Data d => d -> u) -> Ordered a -> u
gmapQ :: forall u. (forall d. Data d => d -> u) -> Ordered a -> [u]
$cgmapQ :: forall a u.
Data a =>
(forall d. Data d => d -> u) -> Ordered a -> [u]
gmapQr :: forall r r'.
(r' -> r -> r)
-> r -> (forall d. Data d => d -> r') -> Ordered a -> r
$cgmapQr :: forall a r r'.
Data a =>
(r' -> r -> r)
-> r -> (forall d. Data d => d -> r') -> Ordered a -> r
gmapQl :: forall r r'.
(r -> r' -> r)
-> r -> (forall d. Data d => d -> r') -> Ordered a -> r
$cgmapQl :: forall a r r'.
Data a =>
(r -> r' -> r)
-> r -> (forall d. Data d => d -> r') -> Ordered a -> r
gmapT :: (forall b. Data b => b -> b) -> Ordered a -> Ordered a
$cgmapT :: forall a.
Data a =>
(forall b. Data b => b -> b) -> Ordered a -> Ordered a
dataCast2 :: forall (t :: * -> * -> *) (c :: * -> *).
Typeable t =>
(forall d e. (Data d, Data e) => c (t d e))
-> Maybe (c (Ordered a))
$cdataCast2 :: forall a (t :: * -> * -> *) (c :: * -> *).
(Data a, Typeable t) =>
(forall d e. (Data d, Data e) => c (t d e))
-> Maybe (c (Ordered a))
dataCast1 :: forall (t :: * -> *) (c :: * -> *).
Typeable t =>
(forall d. Data d => c (t d)) -> Maybe (c (Ordered a))
$cdataCast1 :: forall a (t :: * -> *) (c :: * -> *).
(Data a, Typeable t) =>
(forall d. Data d => c (t d)) -> Maybe (c (Ordered a))
dataTypeOf :: Ordered a -> DataType
$cdataTypeOf :: forall a. Data a => Ordered a -> DataType
toConstr :: Ordered a -> Constr
$ctoConstr :: forall a. Data a => Ordered a -> Constr
gunfold :: forall (c :: * -> *).
(forall b r. Data b => c (b -> r) -> c r)
-> (forall r. r -> c r) -> Constr -> c (Ordered a)
$cgunfold :: forall a (c :: * -> *).
Data a =>
(forall b r. Data b => c (b -> r) -> c r)
-> (forall r. r -> c r) -> Constr -> c (Ordered a)
gfoldl :: forall (c :: * -> *).
(forall d b. Data d => c (d -> b) -> d -> c b)
-> (forall g. g -> c g) -> Ordered a -> c (Ordered a)
$cgfoldl :: forall a (c :: * -> *).
Data a =>
(forall d b. Data d => c (d -> b) -> d -> c b)
-> (forall g. g -> c g) -> Ordered a -> c (Ordered a)
Data, Typeable, forall a.
(forall x. a -> Rep a x) -> (forall x. Rep a x -> a) -> Generic a
forall a x. Rep (Ordered a) x -> Ordered a
forall a x. Ordered a -> Rep (Ordered a) x
$cto :: forall a x. Rep (Ordered a) x -> Ordered a
$cfrom :: forall a x. Ordered a -> Rep (Ordered a) x
Generic, forall a b. a -> Ordered b -> Ordered a
forall a b. (a -> b) -> Ordered a -> Ordered b
forall (f :: * -> *).
(forall a b. (a -> b) -> f a -> f b)
-> (forall a b. a -> f b -> f a) -> Functor f
<$ :: forall a b. a -> Ordered b -> Ordered a
$c<$ :: forall a b. a -> Ordered b -> Ordered a
fmap :: forall a b. (a -> b) -> Ordered a -> Ordered b
$cfmap :: forall a b. (a -> b) -> Ordered a -> Ordered b
Functor, forall a. Eq a => a -> Ordered a -> Bool
forall a. Num a => Ordered a -> a
forall a. Ord a => Ordered a -> a
forall m. Monoid m => Ordered m -> m
forall a. Ordered a -> Bool
forall a. Ordered a -> Int
forall a. Ordered a -> [a]
forall a. (a -> a -> a) -> Ordered a -> a
forall m a. Monoid m => (a -> m) -> Ordered a -> m
forall b a. (b -> a -> b) -> b -> Ordered a -> b
forall a b. (a -> b -> b) -> b -> Ordered a -> b
forall (t :: * -> *).
(forall m. Monoid m => t m -> m)
-> (forall m a. Monoid m => (a -> m) -> t a -> m)
-> (forall m a. Monoid m => (a -> m) -> t a -> m)
-> (forall a b. (a -> b -> b) -> b -> t a -> b)
-> (forall a b. (a -> b -> b) -> b -> t a -> b)
-> (forall b a. (b -> a -> b) -> b -> t a -> b)
-> (forall b a. (b -> a -> b) -> b -> t a -> b)
-> (forall a. (a -> a -> a) -> t a -> a)
-> (forall a. (a -> a -> a) -> t a -> a)
-> (forall a. t a -> [a])
-> (forall a. t a -> Bool)
-> (forall a. t a -> Int)
-> (forall a. Eq a => a -> t a -> Bool)
-> (forall a. Ord a => t a -> a)
-> (forall a. Ord a => t a -> a)
-> (forall a. Num a => t a -> a)
-> (forall a. Num a => t a -> a)
-> Foldable t
product :: forall a. Num a => Ordered a -> a
$cproduct :: forall a. Num a => Ordered a -> a
sum :: forall a. Num a => Ordered a -> a
$csum :: forall a. Num a => Ordered a -> a
minimum :: forall a. Ord a => Ordered a -> a
$cminimum :: forall a. Ord a => Ordered a -> a
maximum :: forall a. Ord a => Ordered a -> a
$cmaximum :: forall a. Ord a => Ordered a -> a
elem :: forall a. Eq a => a -> Ordered a -> Bool
$celem :: forall a. Eq a => a -> Ordered a -> Bool
length :: forall a. Ordered a -> Int
$clength :: forall a. Ordered a -> Int
null :: forall a. Ordered a -> Bool
$cnull :: forall a. Ordered a -> Bool
toList :: forall a. Ordered a -> [a]
$ctoList :: forall a. Ordered a -> [a]
foldl1 :: forall a. (a -> a -> a) -> Ordered a -> a
$cfoldl1 :: forall a. (a -> a -> a) -> Ordered a -> a
foldr1 :: forall a. (a -> a -> a) -> Ordered a -> a
$cfoldr1 :: forall a. (a -> a -> a) -> Ordered a -> a
foldl' :: forall b a. (b -> a -> b) -> b -> Ordered a -> b
$cfoldl' :: forall b a. (b -> a -> b) -> b -> Ordered a -> b
foldl :: forall b a. (b -> a -> b) -> b -> Ordered a -> b
$cfoldl :: forall b a. (b -> a -> b) -> b -> Ordered a -> b
foldr' :: forall a b. (a -> b -> b) -> b -> Ordered a -> b
$cfoldr' :: forall a b. (a -> b -> b) -> b -> Ordered a -> b
foldr :: forall a b. (a -> b -> b) -> b -> Ordered a -> b
$cfoldr :: forall a b. (a -> b -> b) -> b -> Ordered a -> b
foldMap' :: forall m a. Monoid m => (a -> m) -> Ordered a -> m
$cfoldMap' :: forall m a. Monoid m => (a -> m) -> Ordered a -> m
foldMap :: forall m a. Monoid m => (a -> m) -> Ordered a -> m
$cfoldMap :: forall m a. Monoid m => (a -> m) -> Ordered a -> m
fold :: forall m. Monoid m => Ordered m -> m
$cfold :: forall m. Monoid m => Ordered m -> m
Foldable, Functor Ordered
Foldable Ordered
forall (t :: * -> *).
Functor t
-> Foldable t
-> (forall (f :: * -> *) a b.
Applicative f =>
(a -> f b) -> t a -> f (t b))
-> (forall (f :: * -> *) a. Applicative f => t (f a) -> f (t a))
-> (forall (m :: * -> *) a b.
Monad m =>
(a -> m b) -> t a -> m (t b))
-> (forall (m :: * -> *) a. Monad m => t (m a) -> m (t a))
-> Traversable t
forall (m :: * -> *) a. Monad m => Ordered (m a) -> m (Ordered a)
forall (f :: * -> *) a.
Applicative f =>
Ordered (f a) -> f (Ordered a)
forall (m :: * -> *) a b.
Monad m =>
(a -> m b) -> Ordered a -> m (Ordered b)
forall (f :: * -> *) a b.
Applicative f =>
(a -> f b) -> Ordered a -> f (Ordered b)
sequence :: forall (m :: * -> *) a. Monad m => Ordered (m a) -> m (Ordered a)
$csequence :: forall (m :: * -> *) a. Monad m => Ordered (m a) -> m (Ordered a)
mapM :: forall (m :: * -> *) a b.
Monad m =>
(a -> m b) -> Ordered a -> m (Ordered b)
$cmapM :: forall (m :: * -> *) a b.
Monad m =>
(a -> m b) -> Ordered a -> m (Ordered b)
sequenceA :: forall (f :: * -> *) a.
Applicative f =>
Ordered (f a) -> f (Ordered a)
$csequenceA :: forall (f :: * -> *) a.
Applicative f =>
Ordered (f a) -> f (Ordered a)
traverse :: forall (f :: * -> *) a b.
Applicative f =>
(a -> f b) -> Ordered a -> f (Ordered b)
$ctraverse :: forall (f :: * -> *) a b.
Applicative f =>
(a -> f b) -> Ordered a -> f (Ordered b)
Traversable
, forall a. Rep1 Ordered a -> Ordered a
forall a. Ordered a -> Rep1 Ordered a
forall k (f :: k -> *).
(forall (a :: k). f a -> Rep1 f a)
-> (forall (a :: k). Rep1 f a -> f a) -> Generic1 f
$cto1 :: forall a. Rep1 Ordered a -> Ordered a
$cfrom1 :: forall a. Ordered a -> Rep1 Ordered a
Generic1
)
instance Applicative Ordered where
pure :: forall a. a -> Ordered a
pure = forall (m :: * -> *) a. Monad m => a -> m a
return
<*> :: forall a b. Ordered (a -> b) -> Ordered a -> Ordered b
(<*>) = forall (m :: * -> *) a b. Monad m => m (a -> b) -> m a -> m b
ap
instance Monad Ordered where
return :: forall a. a -> Ordered a
return = forall a. a -> Ordered a
Ordered
Ordered a
x >>= :: forall a b. Ordered a -> (a -> Ordered b) -> Ordered b
>>= a -> Ordered b
f = a -> Ordered b
f a
x
instance NFData a => NFData (Ordered a) where
rnf :: Ordered a -> ()
rnf (Ordered a
a) = forall a. NFData a => a -> ()
rnf a
a
instance Hashable a => Hashable (Ordered a)
instance Ord a => Lattice (Ordered a) where
Ordered a
x \/ :: Ordered a -> Ordered a -> Ordered a
\/ Ordered a
y = forall a. a -> Ordered a
Ordered (forall a. Ord a => a -> a -> a
max a
x a
y)
Ordered a
x /\ :: Ordered a -> Ordered a -> Ordered a
/\ Ordered a
y = forall a. a -> Ordered a
Ordered (forall a. Ord a => a -> a -> a
min a
x a
y)
instance (Ord a, Bounded a) => BoundedJoinSemiLattice (Ordered a) where
bottom :: Ordered a
bottom = forall a. a -> Ordered a
Ordered forall a. Bounded a => a
minBound
instance (Ord a, Bounded a) => BoundedMeetSemiLattice (Ordered a) where
top :: Ordered a
top = forall a. a -> Ordered a
Ordered forall a. Bounded a => a
maxBound
instance (Ord a, Bounded a) => Heyting (Ordered a) where
Ordered a
x ==> :: Ordered a -> Ordered a -> Ordered a
==> Ordered a
y | Ordered a
x forall a. Ord a => a -> a -> Bool
> Ordered a
y = Ordered a
y
| Bool
otherwise = forall a. BoundedMeetSemiLattice a => a
top
instance Ord a => PartialOrd (Ordered a) where
leq :: Ordered a -> Ordered a -> Bool
leq = forall a. Ord a => a -> a -> Bool
(<=)
comparable :: Ordered a -> Ordered a -> Bool
comparable Ordered a
_ Ordered a
_ = Bool
True
instance Universe a => Universe (Ordered a) where
universe :: [Ordered a]
universe = forall a b. (a -> b) -> [a] -> [b]
map forall a. a -> Ordered a
Ordered forall a. Universe a => [a]
universe
instance Finite a => Finite (Ordered a) where
universeF :: [Ordered a]
universeF = forall a b. (a -> b) -> [a] -> [b]
map forall a. a -> Ordered a
Ordered forall a. Finite a => [a]
universeF
cardinality :: Tagged (Ordered a) Natural
cardinality = forall {k1} {k2} (s :: k1) b (t :: k2). Tagged s b -> Tagged t b
retag (forall a. Finite a => Tagged a Natural
cardinality :: Tagged a Natural)
instance QC.Arbitrary a => QC.Arbitrary (Ordered a) where
arbitrary :: Gen (Ordered a)
arbitrary = forall a. a -> Ordered a
Ordered forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> forall a. Arbitrary a => Gen a
QC.arbitrary
shrink :: Ordered a -> [Ordered a]
shrink = forall a b. Arbitrary a => (a -> b) -> (b -> a) -> b -> [b]
QC.shrinkMap forall a. a -> Ordered a
Ordered forall a. Ordered a -> a
getOrdered
instance QC.CoArbitrary a => QC.CoArbitrary (Ordered a) where
coarbitrary :: forall b. Ordered a -> Gen b -> Gen b
coarbitrary = forall a b. CoArbitrary a => a -> Gen b -> Gen b
QC.coarbitrary forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. Ordered a -> a
getOrdered
instance QC.Function a => QC.Function (Ordered a) where
function :: forall b. (Ordered a -> b) -> Ordered a :-> b
function = forall b a c.
Function b =>
(a -> b) -> (b -> a) -> (a -> c) -> a :-> c
QC.functionMap forall a. Ordered a -> a
getOrdered forall a. a -> Ordered a
Ordered