Algebra.Heyting.Free

Documentation

data Free a

Free Heyting algebra.

Note: Eq and PartialOrd instances aren't structural.

>>> Top == (Var 'x' ==> Var 'x')
True

>>> Var 'x' == Var 'y'
False

You can test for taulogogies:

>>> leq Top $ (Var 'A' /\ Var 'B' ==> Var 'C') <=>  (Var 'A' ==> Var 'B' ==> Var 'C')
True

>>> leq Top $ (Var 'A' /\ neg (Var 'A')) <=> Bottom
True

>>> leq Top $ (Var 'A' \/ neg (Var 'A')) <=> Top
False

Constructors

Var a

Bottom

Top

(Free a) :/\: (Free a) infixr 6

(Free a) :\/: (Free a) infixr 5

(Free a) :=>: (Free a) infixr 4

Instances

Monad Free

Defined in Algebra.Heyting.Free

Methods

(>>=) :: Free a -> (a -> Free b) -> Free b

(>>) :: Free a -> Free b -> Free b

return :: a -> Free a

fail :: String -> Free a

Functor Free

Defined in Algebra.Heyting.Free

Methods

fmap :: (a -> b) -> Free a -> Free b

(<$) :: a -> Free b -> Free a

Applicative Free

Defined in Algebra.Heyting.Free

Methods

pure :: a -> Free a

(<*>) :: Free (a -> b) -> Free a -> Free b

liftA2 :: (a -> b -> c) -> Free a -> Free b -> Free c

(*>) :: Free a -> Free b -> Free b

(<*) :: Free a -> Free b -> Free a

Foldable Free

Defined in Algebra.Heyting.Free

Methods

fold :: Monoid m => Free m -> m

foldMap :: Monoid m => (a -> m) -> Free a -> m

foldr :: (a -> b -> b) -> b -> Free a -> b

foldr' :: (a -> b -> b) -> b -> Free a -> b

foldl :: (b -> a -> b) -> b -> Free a -> b

foldl' :: (b -> a -> b) -> b -> Free a -> b

foldr1 :: (a -> a -> a) -> Free a -> a

foldl1 :: (a -> a -> a) -> Free a -> a

toList :: Free a -> [a]

null :: Free a -> Bool

length :: Free a -> Int

elem :: Eq a => a -> Free a -> Bool

maximum :: Ord a => Free a -> a

minimum :: Ord a => Free a -> a

sum :: Num a => Free a -> a

product :: Num a => Free a -> a

Traversable Free

Defined in Algebra.Heyting.Free

Methods

traverse :: Applicative f => (a -> f b) -> Free a -> f (Free b)

sequenceA :: Applicative f => Free (f a) -> f (Free a)

mapM :: Monad m => (a -> m b) -> Free a -> m (Free b)

sequence :: Monad m => Free (m a) -> m (Free a)

Ord a => Eq (Free a)

Defined in Algebra.Heyting.Free

Methods

(==) :: Free a -> Free a -> Bool

(/=) :: Free a -> Free a -> Bool

Data a => Data (Free a)

Defined in Algebra.Heyting.Free

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Free a -> c (Free a)

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Free a)

toConstr :: Free a -> Constr

dataTypeOf :: Free a -> DataType

dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (Free a))

dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Free a))

gmapT :: (forall b. Data b => b -> b) -> Free a -> Free a

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Free a -> r

gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Free a -> r

gmapQ :: (forall d. Data d => d -> u) -> Free a -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> Free a -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> Free a -> m (Free a)

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Free a -> m (Free a)

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Free a -> m (Free a)

Show a => Show (Free a)

Defined in Algebra.Heyting.Free

Methods

showsPrec :: Int -> Free a -> ShowS

show :: Free a -> String

showList :: [Free a] -> ShowS

Generic (Free a)

Defined in Algebra.Heyting.Free

Associated Types

type Rep (Free a) :: Type -> Type

Methods

from :: Free a -> Rep (Free a) x

to :: Rep (Free a) x -> Free a

Arbitrary a => Arbitrary (Free a)

Defined in Algebra.Heyting.Free

Methods

arbitrary :: Gen (Free a)

shrink :: Free a -> [Free a]

Ord a => PartialOrd (Free a)

Defined in Algebra.Heyting.Free

Methods

leq :: Free a -> Free a -> Bool

comparable :: Free a -> Free a -> Bool

BoundedMeetSemiLattice (Free a)

Defined in Algebra.Heyting.Free

Methods

top :: Free a

BoundedJoinSemiLattice (Free a)

Defined in Algebra.Heyting.Free

Methods

bottom :: Free a

Lattice (Free a)

Defined in Algebra.Heyting.Free

Methods

(\/) :: Free a -> Free a -> Free a

(/\) :: Free a -> Free a -> Free a

Heyting (Free a)

Defined in Algebra.Heyting.Free

Methods

(==>) :: Free a -> Free a -> Free a

neg :: Free a -> Free a

(<=>) :: Free a -> Free a -> Free a

Generic1 Free

Defined in Algebra.Heyting.Free

Associated Types

type Rep1 Free :: k -> Type

Methods

from1 :: Free a -> Rep1 Free a

to1 :: Rep1 Free a -> Free a

type Rep (Free a)

Defined in Algebra.Heyting.Free

type Rep (Free a) = D1 (MetaData "Free" "Algebra.Heyting.Free" "lattices-2.0.1-DhjhXTEEVqrG2Ss9wckwFx" False) ((C1 (MetaCons "Var" PrefixI False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 a)) :+: (C1 (MetaCons "Bottom" PrefixI False) (U1 :: Type -> Type) :+: C1 (MetaCons "Top" PrefixI False) (U1 :: Type -> Type))) :+: (C1 (MetaCons ":/\\:" (InfixI RightAssociative 6) False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 (Free a)) :*: S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 (Free a))) :+: (C1 (MetaCons ":\\/:" (InfixI RightAssociative 5) False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 (Free a)) :*: S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 (Free a))) :+: C1 (MetaCons ":=>:" (InfixI RightAssociative 4) False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 (Free a)) :*: S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 (Free a))))))

type Rep1 Free

Defined in Algebra.Heyting.Free

type Rep1 Free = D1 (MetaData "Free" "Algebra.Heyting.Free" "lattices-2.0.1-DhjhXTEEVqrG2Ss9wckwFx" False) ((C1 (MetaCons "Var" PrefixI False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) Par1) :+: (C1 (MetaCons "Bottom" PrefixI False) (U1 :: Type -> Type) :+: C1 (MetaCons "Top" PrefixI False) (U1 :: Type -> Type))) :+: (C1 (MetaCons ":/\\:" (InfixI RightAssociative 6) False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec1 Free) :*: S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec1 Free)) :+: (C1 (MetaCons ":\\/:" (InfixI RightAssociative 5) False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec1 Free) :*: S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec1 Free)) :+: C1 (MetaCons ":=>:" (InfixI RightAssociative 4) False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec1 Free) :*: S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec1 Free)))))

liftFree :: a -> Free a

lowerFree :: Heyting b => (a -> b) -> Free a -> b

retractFree :: Heyting a => Free a -> a

substFree :: Free a -> (a -> Free b) -> Free b

toExpr :: Free a -> Expr a