-- Hoogle documentation, generated by Haddock
-- See Hoogle, http://www.haskell.org/hoogle/


-- | L-Fuzzy Set Theory implementation in Haskell
--   
--   If X is a collection of objects denoted generically by x, then a fuzzy
--   set F(A) in X is a set of ordered pairs. Each of them consists of an
--   element x and a membership function which maps x to the membership
--   space M. The current implementation is inspired by the work of Goguen,
--   Joseph A. "L-fuzzy sets." Journal of mathematical analysis and
--   applications 18.1 (1967).
@package lfst
@version 1.0.2


-- | Membership types for the Fuzzy Set definition
module Algebra.LFST.Membership

-- | Membership value between 0 and 1 with Godel join and meet operators
newtype GodelMembership
Godel :: Double -> GodelMembership

-- | Membership value between 0 and 1 with Goguen join and meet operators
newtype GoguenMembership
Goguen :: Double -> GoguenMembership

-- | Membership value between 0 and 1 with Lukasiewicz join and meet
--   operators
newtype LukasiewiczMembership
Lukas :: Double -> LukasiewiczMembership
instance GHC.Num.Num Algebra.LFST.Membership.LukasiewiczMembership
instance GHC.Classes.Ord Algebra.LFST.Membership.LukasiewiczMembership
instance GHC.Classes.Eq Algebra.LFST.Membership.LukasiewiczMembership
instance GHC.Show.Show Algebra.LFST.Membership.LukasiewiczMembership
instance GHC.Num.Num Algebra.LFST.Membership.GoguenMembership
instance GHC.Classes.Ord Algebra.LFST.Membership.GoguenMembership
instance GHC.Classes.Eq Algebra.LFST.Membership.GoguenMembership
instance GHC.Show.Show Algebra.LFST.Membership.GoguenMembership
instance GHC.Num.Num Algebra.LFST.Membership.GodelMembership
instance GHC.Classes.Ord Algebra.LFST.Membership.GodelMembership
instance GHC.Classes.Eq Algebra.LFST.Membership.GodelMembership
instance GHC.Show.Show Algebra.LFST.Membership.GodelMembership
instance Algebra.Lattice.JoinSemiLattice Algebra.LFST.Membership.GodelMembership
instance Algebra.Lattice.MeetSemiLattice Algebra.LFST.Membership.GodelMembership
instance Algebra.Lattice.Lattice Algebra.LFST.Membership.GodelMembership
instance Algebra.Lattice.BoundedJoinSemiLattice Algebra.LFST.Membership.GodelMembership
instance Algebra.Lattice.BoundedMeetSemiLattice Algebra.LFST.Membership.GodelMembership
instance Algebra.Lattice.BoundedLattice Algebra.LFST.Membership.GodelMembership
instance Algebra.Lattice.JoinSemiLattice Algebra.LFST.Membership.GoguenMembership
instance Algebra.Lattice.MeetSemiLattice Algebra.LFST.Membership.GoguenMembership
instance Algebra.Lattice.Lattice Algebra.LFST.Membership.GoguenMembership
instance Algebra.Lattice.BoundedJoinSemiLattice Algebra.LFST.Membership.GoguenMembership
instance Algebra.Lattice.BoundedMeetSemiLattice Algebra.LFST.Membership.GoguenMembership
instance Algebra.Lattice.BoundedLattice Algebra.LFST.Membership.GoguenMembership
instance Algebra.Lattice.JoinSemiLattice Algebra.LFST.Membership.LukasiewiczMembership
instance Algebra.Lattice.MeetSemiLattice Algebra.LFST.Membership.LukasiewiczMembership
instance Algebra.Lattice.Lattice Algebra.LFST.Membership.LukasiewiczMembership
instance Algebra.Lattice.BoundedJoinSemiLattice Algebra.LFST.Membership.LukasiewiczMembership
instance Algebra.Lattice.BoundedMeetSemiLattice Algebra.LFST.Membership.LukasiewiczMembership
instance Algebra.Lattice.BoundedLattice Algebra.LFST.Membership.LukasiewiczMembership


-- | If X is a collection of objects denoted generically by x, then a fuzzy
--   set F(A) in X is a set of ordered pairs. Each of them consists of an
--   element x and a membership function which maps x to the membership
--   space M.
module Algebra.LFST.FuzzySet

-- | FuzzySet type definition
newtype FuzzySet m i
FS :: (Map i m) -> FuzzySet m i

-- | Returns the preimage of the given set in input
preimage :: (Eq i, Eq j) => (i -> j) -> j -> [i] -> [i]

-- | Returns an empty fuzzy set
empty :: (Ord i, BoundedLattice m) => FuzzySet m i

-- | Inserts a new pair (i, m) to the fuzzy set
add :: (Ord i, Eq m, BoundedLattice m) => FuzzySet m i -> (i, m) -> FuzzySet m i

-- | Returns the fuzzy set's support
support :: (Ord i, BoundedLattice m) => FuzzySet m i -> [i]

-- | Returns the element i's membership if i belongs to the support returns
--   its membership, otherwise returns bottom lattice value
mu :: (Ord i, BoundedLattice m) => FuzzySet m i -> i -> m

-- | Returns the crisp subset of given fuzzy set consisting of all elements
--   with membership equals to one
core :: (Ord i, Eq m, BoundedLattice m) => FuzzySet m i -> [i]

-- | Returns those elements whose memberships are greater or equal than the
--   given alpha
alphaCut :: (Ord i, Ord m, BoundedLattice m) => FuzzySet m i -> m -> [i]

-- | Builds a fuzzy set from a list of pairs
fromList :: (Ord i, Eq m, BoundedLattice m) => [(i, m)] -> FuzzySet m i

-- | Applies a unary function to the specified fuzzy set
map1 :: (Ord i, Eq m, BoundedLattice m) => (m -> m) -> FuzzySet m i -> FuzzySet m i

-- | Applies a binary function to the two specified fuzzy sets
map2 :: (Ord i, Eq m, BoundedLattice m) => (m -> m -> m) -> FuzzySet m i -> FuzzySet m i -> FuzzySet m i

-- | Returns the union between the two specified fuzzy sets
union :: (Ord i, Eq m, BoundedLattice m) => FuzzySet m i -> FuzzySet m i -> FuzzySet m i

-- | Returns the intersection between the two specified fuzzy sets
intersection :: (Ord i, Eq m, BoundedLattice m) => FuzzySet m i -> FuzzySet m i -> FuzzySet m i

-- | Returns the complement of the specified fuzzy set
complement :: (Ord i, Num m, Eq m, BoundedLattice m) => FuzzySet m i -> FuzzySet m i

-- | Returns the algebraic sum between the two specified fuzzy sets
algebraicSum :: (Ord i, Eq m, Num m, BoundedLattice m) => FuzzySet m i -> FuzzySet m i -> FuzzySet m i

-- | Returns the algebraic product between the two specified fuzzy sets
algebraicProduct :: (Ord i, Eq m, Num m, BoundedLattice m) => FuzzySet m i -> FuzzySet m i -> FuzzySet m i

-- | Returns the cartesian product between two fuzzy sets using the
--   specified function
generalizedProduct :: (Ord i, Ord j, Eq m, BoundedLattice m) => (m -> m -> m) -> FuzzySet m i -> FuzzySet m j -> FuzzySet m (i, j)

-- | Defines a mapping between sub-categories preserving morphisms
class ExoFunctor f i where type family SubCatConstraintI f i :: Constraint type family SubCatConstraintJ f j :: Constraint SubCatConstraintI f i = () SubCatConstraintJ f j = ()
fmap :: (ExoFunctor f i, SubCatConstraintI f i, SubCatConstraintJ f j) => (i -> j) -> f i -> f j
instance (GHC.Classes.Ord m, GHC.Classes.Ord i) => GHC.Classes.Ord (Algebra.LFST.FuzzySet.FuzzySet m i)
instance (GHC.Classes.Eq m, GHC.Classes.Eq i) => GHC.Classes.Eq (Algebra.LFST.FuzzySet.FuzzySet m i)
instance (GHC.Classes.Ord i, Algebra.Lattice.BoundedLattice m, GHC.Show.Show i, GHC.Show.Show m) => GHC.Show.Show (Algebra.LFST.FuzzySet.FuzzySet m i)
instance (Algebra.Lattice.BoundedLattice m, GHC.Classes.Eq m) => Algebra.LFST.FuzzySet.ExoFunctor (Algebra.LFST.FuzzySet.FuzzySet m) i