Algebra.Heyting.Free.Expr

Documentation

data Expr a

Heyting algebra expression.

Note: this type doesn't have Heyting instance, as its Eq and Ord are structural.

Constructors

Var a

Bottom

Top

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

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

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

Instances

Monad Expr

Defined in Algebra.Heyting.Free.Expr

Methods

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

(>>) :: Expr a -> Expr b -> Expr b

return :: a -> Expr a

fail :: String -> Expr a

Functor Expr

Defined in Algebra.Heyting.Free.Expr

Methods

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

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

Applicative Expr

Defined in Algebra.Heyting.Free.Expr

Methods

pure :: a -> Expr a

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

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

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

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

Foldable Expr

Defined in Algebra.Heyting.Free.Expr

Methods

fold :: Monoid m => Expr m -> m

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

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

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

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

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

foldr1 :: (a -> a -> a) -> Expr a -> a

foldl1 :: (a -> a -> a) -> Expr a -> a

toList :: Expr a -> [a]

null :: Expr a -> Bool

length :: Expr a -> Int

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

maximum :: Ord a => Expr a -> a

minimum :: Ord a => Expr a -> a

sum :: Num a => Expr a -> a

product :: Num a => Expr a -> a

Traversable Expr

Defined in Algebra.Heyting.Free.Expr

Methods

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

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

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

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

Eq a => Eq (Expr a)

Defined in Algebra.Heyting.Free.Expr

Methods

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

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

Data a => Data (Expr a)

Defined in Algebra.Heyting.Free.Expr

Methods

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

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

toConstr :: Expr a -> Constr

dataTypeOf :: Expr a -> DataType

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

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

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

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

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

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

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

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

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

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

Ord a => Ord (Expr a)

Defined in Algebra.Heyting.Free.Expr

Methods

compare :: Expr a -> Expr a -> Ordering

(<) :: Expr a -> Expr a -> Bool

(<=) :: Expr a -> Expr a -> Bool

(>) :: Expr a -> Expr a -> Bool

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

max :: Expr a -> Expr a -> Expr a

min :: Expr a -> Expr a -> Expr a

Show a => Show (Expr a)

Defined in Algebra.Heyting.Free.Expr

Methods

showsPrec :: Int -> Expr a -> ShowS

show :: Expr a -> String

showList :: [Expr a] -> ShowS

Generic (Expr a)

Defined in Algebra.Heyting.Free.Expr

Associated Types

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

Methods

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

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

Generic1 Expr

Defined in Algebra.Heyting.Free.Expr

Associated Types

type Rep1 Expr :: k -> Type

Methods

from1 :: Expr a -> Rep1 Expr a

to1 :: Rep1 Expr a -> Expr a

type Rep (Expr a)

Defined in Algebra.Heyting.Free.Expr

type Rep (Expr a) = D1 (MetaData "Expr" "Algebra.Heyting.Free.Expr" "lattices-2.0.2-HdMTcqWeXqlAAQvdNaFFrQ" 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 (Expr a)) :*: S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 (Expr a))) :+: (C1 (MetaCons ":\\/:" (InfixI RightAssociative 5) False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 (Expr a)) :*: S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 (Expr a))) :+: C1 (MetaCons ":=>:" (InfixI RightAssociative 4) False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 (Expr a)) :*: S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 (Expr a))))))

type Rep1 Expr

Defined in Algebra.Heyting.Free.Expr

type Rep1 Expr = D1 (MetaData "Expr" "Algebra.Heyting.Free.Expr" "lattices-2.0.2-HdMTcqWeXqlAAQvdNaFFrQ" 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 Expr) :*: S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec1 Expr)) :+: (C1 (MetaCons ":\\/:" (InfixI RightAssociative 5) False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec1 Expr) :*: S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec1 Expr)) :+: C1 (MetaCons ":=>:" (InfixI RightAssociative 4) False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec1 Expr) :*: S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec1 Expr)))))

proofSearch :: forall a. Ord a => Expr a -> Bool

Decide whether x :: Expr a is provable.

Note: this doesn't construct a proof term, but merely returns a Bool.