{-# LANGUAGE DeriveDataTypeable  #-}
{-# LANGUAGE DeriveFoldable      #-}
{-# LANGUAGE DeriveFunctor       #-}
{-# LANGUAGE DeriveGeneric       #-}
{-# LANGUAGE DeriveTraversable   #-}
{-# LANGUAGE FlexibleContexts    #-}
{-# LANGUAGE Safe                #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE TypeOperators       #-}
module Algebra.Lattice.Stacked (
    Stacked(..)
  , foldStacked
  ) where

import Prelude ()
import Prelude.Compat

import Algebra.Lattice
import Algebra.PartialOrd

import Control.DeepSeq       (NFData (..))
import Control.Monad         (ap, liftM2)
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

--
-- Stacked
--

-- | Stacked two lattices, one on top of another. All minimal elements of upper lattice cover all maximal elements of lower lattice.
data Stacked a b = Lower a
              | Upper b
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forall a (m :: * -> *) a.
Monad m =>
Stacked a (m a) -> m (Stacked a a)
forall a (f :: * -> *) a.
Applicative f =>
Stacked a (f a) -> f (Stacked a a)
forall a (m :: * -> *) a b.
Monad m =>
(a -> m b) -> Stacked a a -> m (Stacked a b)
forall a (f :: * -> *) a b.
Applicative f =>
(a -> f b) -> Stacked a a -> f (Stacked a b)
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 =>
Stacked a (m a) -> m (Stacked a a)
forall (f :: * -> *) a.
Applicative f =>
Stacked a (f a) -> f (Stacked a a)
forall (m :: * -> *) a b.
Monad m =>
(a -> m b) -> Stacked a a -> m (Stacked a b)
forall (f :: * -> *) a b.
Applicative f =>
(a -> f b) -> Stacked a a -> f (Stacked a b)
sequence :: Stacked a (m a) -> m (Stacked a a)
$csequence :: forall a (m :: * -> *) a.
Monad m =>
Stacked a (m a) -> m (Stacked a a)
mapM :: (a -> m b) -> Stacked a a -> m (Stacked a b)
$cmapM :: forall a (m :: * -> *) a b.
Monad m =>
(a -> m b) -> Stacked a a -> m (Stacked a b)
sequenceA :: Stacked a (f a) -> f (Stacked a a)
$csequenceA :: forall a (f :: * -> *) a.
Applicative f =>
Stacked a (f a) -> f (Stacked a a)
traverse :: (a -> f b) -> Stacked a a -> f (Stacked a b)
$ctraverse :: forall a (f :: * -> *) a b.
Applicative f =>
(a -> f b) -> Stacked a a -> f (Stacked a b)
$cp2Traversable :: forall a. Foldable (Stacked a)
$cp1Traversable :: forall a. Functor (Stacked a)
Traversable
            , (forall a. Stacked a a -> Rep1 (Stacked a) a)
-> (forall a. Rep1 (Stacked a) a -> Stacked a a)
-> Generic1 (Stacked a)
forall a. Rep1 (Stacked a) a -> Stacked a a
forall a. Stacked a a -> Rep1 (Stacked a) a
forall a a. Rep1 (Stacked a) a -> Stacked a a
forall a a. Stacked a a -> Rep1 (Stacked a) 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 a. Rep1 (Stacked a) a -> Stacked a a
$cfrom1 :: forall a a. Stacked a a -> Rep1 (Stacked a) a
Generic1
            )

foldStacked :: (l -> r) -> (u -> r) -> Stacked l u -> r
foldStacked :: (l -> r) -> (u -> r) -> Stacked l u -> r
foldStacked l -> r
f u -> r
_ (Lower l
l) = l -> r
f l
l
foldStacked l -> r
_ u -> r
f (Upper u
u) = u -> r
f u
u

instance Applicative (Stacked a) where
    pure :: a -> Stacked a a
pure = a -> Stacked a a
forall a a. a -> Stacked a a
Upper
    <*> :: Stacked a (a -> b) -> Stacked a a -> Stacked a b
(<*>) = Stacked a (a -> b) -> Stacked a a -> Stacked a b
forall (m :: * -> *) a b. Monad m => m (a -> b) -> m a -> m b
ap

instance Monad (Stacked a) where
    return :: a -> Stacked a a
return = a -> Stacked a a
forall (f :: * -> *) a. Applicative f => a -> f a
pure

    Lower a
x >>= :: Stacked a a -> (a -> Stacked a b) -> Stacked a b
>>= a -> Stacked a b
_ = a -> Stacked a b
forall a b. a -> Stacked a b
Lower a
x
    Upper a
x >>= a -> Stacked a b
f = a -> Stacked a b
f a
x

instance (NFData a, NFData b) => NFData (Stacked a b) where
    rnf :: Stacked a b -> ()
rnf (Upper b
x) = b -> ()
forall a. NFData a => a -> ()
rnf b
x
    rnf (Lower a
x) = a -> ()
forall a. NFData a => a -> ()
rnf a
x

instance (Hashable a, Hashable b) => Hashable (Stacked a b)

instance (PartialOrd a, PartialOrd b) => PartialOrd (Stacked a b) where
    leq :: Stacked a b -> Stacked a b -> Bool
leq (Upper b
x) (Upper b
y) = b -> b -> Bool
forall a. PartialOrd a => a -> a -> Bool
leq b
x b
y
    leq (Upper b
_) Stacked a b
_ = Bool
False
    leq Stacked a b
_ (Upper b
_) = Bool
True
    leq (Lower a
x) (Lower a
y) = a -> a -> Bool
forall a. PartialOrd a => a -> a -> Bool
leq a
x a
y
    comparable :: Stacked a b -> Stacked a b -> Bool
comparable (Upper b
x) (Upper b
y) = b -> b -> Bool
forall a. PartialOrd a => a -> a -> Bool
comparable b
x b
y
    comparable (Upper b
_) Stacked a b
_ = Bool
True
    comparable Stacked a b
_ (Upper b
_) = Bool
True
    comparable (Lower a
x) (Lower a
y) = a -> a -> Bool
forall a. PartialOrd a => a -> a -> Bool
comparable a
x a
y

instance (Lattice a, Lattice b) => Lattice (Stacked a b) where
    Upper b
x \/ :: Stacked a b -> Stacked a b -> Stacked a b
\/ Upper b
y = b -> Stacked a b
forall a a. a -> Stacked a a
Upper (b
x b -> b -> b
forall a. Lattice a => a -> a -> a
\/ b
y)
    u :: Stacked a b
u@(Upper b
_) \/ Stacked a b
_ = Stacked a b
u
    Stacked a b
_ \/ u :: Stacked a b
u@(Upper b
_) = Stacked a b
u
    Lower a
x \/ Lower a
y = a -> Stacked a b
forall a b. a -> Stacked a b
Lower (a
x a -> a -> a
forall a. Lattice a => a -> a -> a
\/ a
y)
    Lower a
x /\ :: Stacked a b -> Stacked a b -> Stacked a b
/\ Lower a
y = a -> Stacked a b
forall a b. a -> Stacked a b
Lower (a
x a -> a -> a
forall a. Lattice a => a -> a -> a
/\ a
y)
    l :: Stacked a b
l@(Lower a
_) /\ Stacked a b
_ = Stacked a b
l
    Stacked a b
_ /\ l :: Stacked a b
l@(Lower a
_) = Stacked a b
l
    Upper b
x /\ Upper b
y = b -> Stacked a b
forall a a. a -> Stacked a a
Upper (b
x b -> b -> b
forall a. Lattice a => a -> a -> a
/\ b
y)

instance (BoundedJoinSemiLattice a, Lattice b) => BoundedJoinSemiLattice (Stacked a b) where
    bottom :: Stacked a b
bottom = a -> Stacked a b
forall a b. a -> Stacked a b
Lower a
forall a. BoundedJoinSemiLattice a => a
bottom

instance (Lattice a, BoundedMeetSemiLattice b) => BoundedMeetSemiLattice (Stacked a b) where
    top :: Stacked a b
top = b -> Stacked a b
forall a a. a -> Stacked a a
Upper b
forall a. BoundedMeetSemiLattice a => a
top

instance (Universe a, Universe b) => Universe (Stacked a b) where
    universe :: [Stacked a b]
universe = (a -> Stacked a b
forall a b. a -> Stacked a b
Lower (a -> Stacked a b) -> [a] -> [Stacked a b]
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> [a]
forall a. Universe a => [a]
universe) [Stacked a b] -> [Stacked a b] -> [Stacked a b]
forall a. [a] -> [a] -> [a]
+++ (b -> Stacked a b
forall a a. a -> Stacked a a
Upper (b -> Stacked a b) -> [b] -> [Stacked a b]
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> [b]
forall a. Universe a => [a]
universe)

instance (Finite a, Finite b) => Finite (Stacked a b) where
    universeF :: [Stacked a b]
universeF = (a -> Stacked a b
forall a b. a -> Stacked a b
Lower (a -> Stacked a b) -> [a] -> [Stacked a b]
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> [a]
forall a. Universe a => [a]
universe) [Stacked a b] -> [Stacked a b] -> [Stacked a b]
forall a. [a] -> [a] -> [a]
++ (b -> Stacked a b
forall a a. a -> Stacked a a
Upper (b -> Stacked a b) -> [b] -> [Stacked a b]
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> [b]
forall a. Universe a => [a]
universe)
    cardinality :: Tagged (Stacked a b) Natural
cardinality = (Natural -> Natural -> Natural)
-> Tagged (Stacked a b) Natural
-> Tagged (Stacked a b) Natural
-> Tagged (Stacked a b) Natural
forall (m :: * -> *) a1 a2 r.
Monad m =>
(a1 -> a2 -> r) -> m a1 -> m a2 -> m r
liftM2 Natural -> Natural -> Natural
forall a. Num a => a -> a -> a
(+)
        (Tagged a Natural -> Tagged (Stacked a b) Natural
forall k1 k2 (s :: k1) b (t :: k2). Tagged s b -> Tagged t b
retag (Tagged a Natural
forall a. Finite a => Tagged a Natural
cardinality :: Tagged a Natural))
        (Tagged b Natural -> Tagged (Stacked a b) Natural
forall k1 k2 (s :: k1) b (t :: k2). Tagged s b -> Tagged t b
retag (Tagged b Natural
forall a. Finite a => Tagged a Natural
cardinality :: Tagged b Natural))

instance (QC.Arbitrary a, QC.Arbitrary b) => QC.Arbitrary (Stacked a b) where
    arbitrary :: Gen (Stacked a b)
arbitrary = [Gen (Stacked a b)] -> Gen (Stacked a b)
forall a. [Gen a] -> Gen a
QC.oneof [b -> Stacked a b
forall a a. a -> Stacked a a
Upper (b -> Stacked a b) -> Gen b -> Gen (Stacked a b)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> Gen b
forall a. Arbitrary a => Gen a
QC.arbitrary, a -> Stacked a b
forall a b. a -> Stacked a b
Lower (a -> Stacked a b) -> Gen a -> Gen (Stacked a b)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> Gen a
forall a. Arbitrary a => Gen a
QC.arbitrary]
    shrink :: Stacked a b -> [Stacked a b]
shrink (Lower a
x) = a -> Stacked a b
forall a b. a -> Stacked a b
Lower (a -> Stacked a b) -> [a] -> [Stacked a b]
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> a -> [a]
forall a. Arbitrary a => a -> [a]
QC.shrink a
x
    shrink (Upper b
y) = b -> Stacked a b
forall a a. a -> Stacked a a
Upper (b -> Stacked a b) -> [b] -> [Stacked a b]
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> b -> [b]
forall a. Arbitrary a => a -> [a]
QC.shrink b
y