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


-- | numerical spaces
--   
--   Spaces as higher-kinded numbers.
@package numhask-space
@version 0.1.1

module NumHask.Analysis.Space

-- | a continuous set of numbers mathematics does not define a space, so
--   library devs are free to experiment.
--   
--   <pre>
--   a `contains` union a b &amp;&amp; b `contains` union a b
--   lower a \/ upper a == lower a
--   lower a /\ upper a == upper a
--   </pre>
class (Spaceable (Element s)) => Space s where {
    
    -- | the underlying element in the space
    type family Element s :: *;
}

-- | lower boundary
lower :: Space s => s -> Element s

-- | upper boundary
upper :: Space s => s -> Element s

-- | space containing a single element
singleton :: Space s => Element s -> s

-- | the intersection of two spaces
intersection :: Space s => s -> s -> s

-- | the union of two spaces
union :: Space s => s -> s -> s

-- | Normalise a space so that &gt; lower a / upper a == lower a &gt; lower
--   a / upper a == upper a
norm :: Space s => s -> s

-- | create a normalised space from two elements
(...) :: Space s => Element s -> Element s -> s

-- | create a space from two elements witjout normalising
(>.<) :: Space s => Element s -> Element s -> s
infix 3 >.<
infix 3 ...
type Spaceable a = (Eq a, JoinSemiLattice a, MeetSemiLattice a)
newtype Union a
Union :: a -> Union a
[getUnion] :: Union a -> a
newtype Intersection a
Intersection :: a -> Intersection a
[getIntersection] :: Intersection a -> a

-- | a space that can be divided neatly
class (Space s, Subtractive (Element s), Field (Element s)) => FieldSpace s where {
    type family Grid s :: *;
}

-- | create equally-spaced elements across a space
grid :: FieldSpace s => Pos -> s -> Grid s -> [Element s]

-- | create equally-spaced spaces from a space
gridSpace :: FieldSpace s => s -> Grid s -> [s]

-- | mid-point of the space
mid :: (Space s, Field (Element s)) => s -> Element s

-- | project a data point from one space to another, preserving relative
--   position
--   
--   <pre>
--   project o n (lower o) = lower n
--   project o n (upper o) = upper n
--   project a a x = x
--   </pre>
project :: (Space s, Field (Element s), Subtractive (Element s)) => s -> s -> Element s -> Element s

-- | Pos suggests where points should be placed in forming a grid across a
--   field space.
data Pos
OuterPos :: Pos
InnerPos :: Pos
LowerPos :: Pos
UpperPos :: Pos
MidPos :: Pos

-- | the containing space of a non-empty Foldable
space1 :: (Space s, Traversable f) => f (Element s) -> s

-- | is an element in the space
(|.|) :: Space s => Element s -> s -> Bool
infixl 7 |.|
memberOf :: Space s => Element s -> s -> Bool

-- | is a space contained within another?
contains :: Space s => s -> s -> Bool

-- | are two spaces disjoint?
disjoint :: Space s => s -> s -> Bool

-- | is one space completely above the other
(|>|) :: Space s => s -> s -> Bool
infixl 7 |>|

-- | is one space completely below the other
(|<|) :: Space s => s -> s -> Bool
infixl 7 |<|

-- | distance between boundaries
width :: (Space s, Subtractive (Element s)) => s -> Element s

-- | create a space centered on a plus or minus b
(+/-) :: (Space s, Subtractive (Element s)) => Element s -> Element s -> s
infixl 6 +/-

-- | lift a monotone function (increasing or decreasing) over a given space
monotone :: (Space a, Space b) => (Element a -> Element b) -> a -> b

-- | a big, big space
whole :: (Space s, BoundedJoinSemiLattice (Element s), BoundedMeetSemiLattice (Element s)) => s

-- | a negative space
negWhole :: (Space s, BoundedJoinSemiLattice (Element s), BoundedMeetSemiLattice (Element s)) => s

-- | a small space
eps :: (Space s, Epsilon (Element s), Multiplicative (Element s)) => Element s -> Element s -> s

-- | widen a space
widen :: (Space s, Subtractive (Element s)) => Element s -> s -> s

-- | widen by a small amount
widenEps :: (Space s, Epsilon (Element s), Multiplicative (Element s)) => Element s -> s -> s
instance GHC.Classes.Eq NumHask.Analysis.Space.Pos
instance GHC.Show.Show NumHask.Analysis.Space.Pos
instance NumHask.Analysis.Space.Space a => GHC.Base.Semigroup (NumHask.Analysis.Space.Intersection a)
instance (NumHask.Algebra.Abstract.Lattice.BoundedMeetSemiLattice a, NumHask.Analysis.Space.Space a) => GHC.Base.Monoid (NumHask.Analysis.Space.Intersection a)
instance NumHask.Analysis.Space.Space a => GHC.Base.Semigroup (NumHask.Analysis.Space.Union a)
instance (NumHask.Algebra.Abstract.Lattice.BoundedJoinSemiLattice a, NumHask.Analysis.Space.Space a) => GHC.Base.Monoid (NumHask.Analysis.Space.Union a)


-- | An Space with no empty, a semigroup based on a convex hull union, and
--   a monoid on a negative space.
module NumHask.Data.Range

-- | A continuous range over type a
--   
--   <pre>
--   &gt;&gt;&gt; let a = Range (-1) 1
--   
--   &gt;&gt;&gt; a
--   Range -1 1
--   
--   &gt;&gt;&gt; fmap (+1) (Range 1 2)
--   Range 2 3
--   
--   &gt;&gt;&gt; one :: Range Double
--   Range -0.5 0.5
--   
--   &gt;&gt;&gt; zero :: Range Double
--   Range Infinity -Infinity
--   </pre>
--   
--   as a Field instance
--   
--   <pre>
--   &gt;&gt;&gt; Range 0 1 + zero
--   Range 0.0 1.0
--   
--   &gt;&gt;&gt; Range 0 1 + Range 2 3
--   Range 0.0 3.0
--   
--   &gt;&gt;&gt; Range 1 1 - one
--   Range 0.5 1.0
--   
--   &gt;&gt;&gt; Range 0 1 * one
--   Range 0.0 1.0
--   
--   &gt;&gt;&gt; Range 0 1 / one
--   Range 0.0 1.0
--   
--   &gt;&gt;&gt; abs (Range 1 0)
--   Range 0.0 1.0
--   
--   &gt;&gt;&gt; sign (Range 1 0) == negate one
--   True
--   </pre>
--   
--   Idempotent
--   
--   <pre>
--   &gt;&gt;&gt; Range 0 2 + Range 0 2
--   Range 0.0 2.0
--   </pre>
--   
--   as a space instance
--   
--   <pre>
--   &gt;&gt;&gt; NumHask.Space.project (Range 0 1) (Range 1 4) 0.5
--   2.5
--   
--   &gt;&gt;&gt; NumHask.Space.grid NumHask.Space.OuterPos (Range 0 10) 5
--   [0.0,2.0,4.0,6.0,8.0,10.0]
--   
--   &gt;&gt;&gt; NumHask.Space.gridSpace (Range 0 1) 4
--   [Range 0.0 0.25,Range 0.25 0.5,Range 0.5 0.75,Range 0.75 1.0]
--   
--   &gt;&gt;&gt; gridSensible NumHask.Space.OuterPos (Range (-12.0) 23.0) 6
--   [-10.0,-5.0,0.0,5.0,10.0,15.0,20.0]
--   </pre>
newtype Range a
Range' :: (a, a) -> Range a

-- | A tuple is the preferred concrete implementation of a Range, due to
--   many libraries having substantial optimizations for tuples already (eg
--   <tt>Vector</tt>). 'Pattern Synonyms' allow us to recover a constructor
--   without the need for tuple syntax.
pattern Range :: a -> a -> Range a
gridSensible :: (Ord a, JoinSemiLattice a, FromInteger a, FromRatio a, QuotientField a Integer, ExpField a, Epsilon a) => Pos -> Bool -> Range a -> Integer -> [a]
instance GHC.Generics.Generic (NumHask.Data.Range.Range a)
instance GHC.Classes.Eq a => GHC.Classes.Eq (NumHask.Data.Range.Range a)
instance GHC.Show.Show a => GHC.Show.Show (NumHask.Data.Range.Range a)
instance Data.Functor.Classes.Eq1 NumHask.Data.Range.Range
instance Data.Functor.Classes.Show1 NumHask.Data.Range.Range
instance GHC.Base.Functor NumHask.Data.Range.Range
instance Data.Functor.Bind.Class.Apply NumHask.Data.Range.Range
instance GHC.Base.Applicative NumHask.Data.Range.Range
instance Data.Foldable.Foldable NumHask.Data.Range.Range
instance Data.Semigroup.Foldable.Class.Foldable1 NumHask.Data.Range.Range
instance Data.Traversable.Traversable NumHask.Data.Range.Range
instance Data.Semigroup.Traversable.Class.Traversable1 NumHask.Data.Range.Range
instance Data.Distributive.Distributive NumHask.Data.Range.Range
instance Data.Functor.Rep.Representable NumHask.Data.Range.Range
instance NumHask.Algebra.Abstract.Lattice.JoinSemiLattice a => NumHask.Algebra.Abstract.Lattice.JoinSemiLattice (NumHask.Data.Range.Range a)
instance NumHask.Algebra.Abstract.Lattice.MeetSemiLattice a => NumHask.Algebra.Abstract.Lattice.MeetSemiLattice (NumHask.Data.Range.Range a)
instance NumHask.Algebra.Abstract.Lattice.BoundedLattice a => NumHask.Algebra.Abstract.Lattice.BoundedJoinSemiLattice (NumHask.Data.Range.Range a)
instance NumHask.Algebra.Abstract.Lattice.BoundedLattice a => NumHask.Algebra.Abstract.Lattice.BoundedMeetSemiLattice (NumHask.Data.Range.Range a)
instance NumHask.Algebra.Abstract.Lattice.Lattice a => NumHask.Analysis.Space.Space (NumHask.Data.Range.Range a)
instance (NumHask.Algebra.Abstract.Lattice.Lattice a, NumHask.Algebra.Abstract.Field.Field a, NumHask.Algebra.Abstract.Additive.Subtractive a, NumHask.Data.Integral.FromInteger a) => NumHask.Analysis.Space.FieldSpace (NumHask.Data.Range.Range a)
instance NumHask.Algebra.Abstract.Lattice.BoundedLattice a => GHC.Base.Semigroup (NumHask.Data.Range.Range a)
instance NumHask.Algebra.Abstract.Lattice.BoundedLattice a => GHC.Base.Monoid (NumHask.Data.Range.Range a)
instance (NumHask.Algebra.Abstract.Additive.Additive a, NumHask.Algebra.Abstract.Lattice.Lattice a) => NumHask.Algebra.Abstract.Additive.Additive (NumHask.Data.Range.Range a)
instance (NumHask.Algebra.Abstract.Additive.Subtractive a, NumHask.Algebra.Abstract.Lattice.Lattice a) => NumHask.Algebra.Abstract.Additive.Subtractive (NumHask.Data.Range.Range a)
instance (NumHask.Algebra.Abstract.Multiplicative.Multiplicative a, NumHask.Algebra.Abstract.Lattice.Lattice a) => NumHask.Algebra.Abstract.Multiplicative.Multiplicative (NumHask.Data.Range.Range a)
instance (NumHask.Algebra.Abstract.Lattice.BoundedLattice a, NumHask.Analysis.Metric.Epsilon a, NumHask.Algebra.Abstract.Multiplicative.Divisive a) => NumHask.Algebra.Abstract.Multiplicative.Divisive (NumHask.Data.Range.Range a)
instance (NumHask.Algebra.Abstract.Multiplicative.Multiplicative a, NumHask.Algebra.Abstract.Additive.Subtractive a, NumHask.Algebra.Abstract.Lattice.Lattice a) => NumHask.Analysis.Metric.Signed (NumHask.Data.Range.Range a)
instance (NumHask.Data.Integral.FromInteger a, NumHask.Algebra.Abstract.Lattice.Lattice a) => NumHask.Data.Integral.FromInteger (NumHask.Data.Range.Range a)
instance NumHask.Algebra.Abstract.Additive.Additive a => NumHask.Algebra.Abstract.Action.AdditiveAction (NumHask.Data.Range.Range a)
instance NumHask.Algebra.Abstract.Additive.Subtractive a => NumHask.Algebra.Abstract.Action.SubtractiveAction (NumHask.Data.Range.Range a)
instance NumHask.Algebra.Abstract.Multiplicative.Multiplicative a => NumHask.Algebra.Abstract.Action.MultiplicativeAction (NumHask.Data.Range.Range a)
instance NumHask.Algebra.Abstract.Multiplicative.Divisive a => NumHask.Algebra.Abstract.Action.DivisiveAction (NumHask.Data.Range.Range a)


-- | representation of a possibly discontinuous interval
module NumHask.Data.RangeD
newtype RangeD a
RangeD :: [Range a] -> RangeD a
normalise :: (Ord a, Lattice a, Subtractive a) => RangeD a -> RangeD a
instance Data.Traversable.Traversable NumHask.Data.RangeD.RangeD
instance Data.Foldable.Foldable NumHask.Data.RangeD.RangeD
instance GHC.Base.Functor NumHask.Data.RangeD.RangeD
instance GHC.Show.Show a => GHC.Show.Show (NumHask.Data.RangeD.RangeD a)
instance GHC.Generics.Generic (NumHask.Data.RangeD.RangeD a)
instance GHC.Classes.Eq a => GHC.Classes.Eq (NumHask.Data.RangeD.RangeD a)
instance (GHC.Classes.Ord a, NumHask.Algebra.Abstract.Lattice.Lattice a, NumHask.Algebra.Abstract.Additive.Subtractive a) => NumHask.Algebra.Abstract.Additive.Additive (NumHask.Data.RangeD.RangeD a)
instance (NumHask.Algebra.Abstract.Multiplicative.Divisive a, GHC.Classes.Ord a, NumHask.Algebra.Abstract.Lattice.Lattice a, NumHask.Algebra.Abstract.Additive.Subtractive a) => NumHask.Algebra.Abstract.Additive.Subtractive (NumHask.Data.RangeD.RangeD a)
instance (GHC.Classes.Ord a, NumHask.Algebra.Abstract.Lattice.Lattice a, NumHask.Algebra.Abstract.Additive.Subtractive a, NumHask.Algebra.Abstract.Multiplicative.Multiplicative a) => NumHask.Algebra.Abstract.Multiplicative.Multiplicative (NumHask.Data.RangeD.RangeD a)
instance (NumHask.Algebra.Abstract.Multiplicative.Multiplicative a, NumHask.Algebra.Abstract.Lattice.BoundedLattice a, NumHask.Analysis.Metric.Epsilon a, GHC.Classes.Ord a, NumHask.Algebra.Abstract.Additive.Subtractive a, NumHask.Algebra.Abstract.Multiplicative.Divisive a) => NumHask.Algebra.Abstract.Multiplicative.Divisive (NumHask.Data.RangeD.RangeD a)


-- | a two-dimensional plane, implemented as a composite of a <a>Pair</a>
--   of <a>Range</a>s.
module NumHask.Data.Rect

-- | a <a>Pair</a> of <a>Ranges</a> that form a rectangle in what is often
--   thought of as the XY plane.
--   
--   <pre>
--   &gt;&gt;&gt; let a = Rect (-1) 1 (-2) 4
--   
--   &gt;&gt;&gt; a
--   Rect -1 1 -2 4
--   
--   &gt;&gt;&gt; let (Ranges x y) = a
--   
--   &gt;&gt;&gt; x
--   Range -1 1
--   
--   &gt;&gt;&gt; y
--   Range -2 4
--   
--   &gt;&gt;&gt; fmap (+1) (Rect 1 2 3 4)
--   Rect 2 3 4 5
--   
--   &gt;&gt;&gt; one :: Rect Double
--   Rect -0.5 0.5 -0.5 0.5
--   
--   &gt;&gt;&gt; zero :: Rect Double
--   Rect Infinity -Infinity Infinity -Infinity
--   </pre>
--   
--   as a Field instance
--   
--   <pre>
--   &gt;&gt;&gt; Rect 0 1 2 3 + zero
--   Rect 0.0 1.0 2.0 3.0
--   
--   &gt;&gt;&gt; Rect 0 1 (-2) (-1) + Rect 2 3 (-5) 3
--   Rect 0.0 3.0 -5.0 3.0
--   
--   &gt;&gt;&gt; Rect 1 1 1 1 - one
--   Rect 0.5 1.0 0.5 1.0
--   
--   &gt;&gt;&gt; Rect 0 1 0 1 * one
--   Rect 0.0 1.0 0.0 1.0
--   
--   &gt;&gt;&gt; Rect 0 1 0 1 / one
--   Rect 0.0 1.0 0.0 1.0
--   
--   &gt;&gt;&gt; singleton (Pair 1.0 2.0) :: Rect Double
--   Rect 1.0 1.0 2.0 2.0
--   
--   &gt;&gt;&gt; abs (Rect 1 0 1 0)
--   Rect 0.0 1.0 0.0 1.0
--   
--   &gt;&gt;&gt; sign (Rect 1 0 1 0) == negate one
--   True
--   </pre>
--   
--   as a Space instance
--   
--   <pre>
--   &gt;&gt;&gt; project (Rect 0 1 (-1) 0) (Rect 1 4 10 0) (Pair 0.5 1)
--   Pair 2.5 -10.0
--   
--   &gt;&gt;&gt; gridSpace (Rect 0 10 0 1) (Pair 2 2)
--   [Rect 0.0 5.0 0.0 0.5,Rect 0.0 5.0 0.5 1.0,Rect 5.0 10.0 0.0 0.5,Rect 5.0 10.0 0.5 1.0]
--   
--   &gt;&gt;&gt; grid MidPos (Rect 0 10 0 1) (Pair 2 2)
--   [Pair 2.5 0.25,Pair 2.5 0.75,Pair 7.5 0.25,Pair 7.5 0.75]
--   </pre>
newtype Rect a
Rect' :: Compose Pair Range a -> Rect a

-- | pattern of Rect lowerx upperx lowery uppery
pattern Rect :: a -> a -> a -> a -> Rect a

-- | pattern of Ranges xrange yrange
pattern Ranges :: Range a -> Range a -> Rect a

-- | create a list of pairs representing the lower left and upper right
--   cormners of a rectangle.
corners :: Lattice a => Rect a -> [Pair a]

-- | project a Rect from an old range to a new one
projectRect :: (Lattice a, Subtractive a, Field a) => Rect a -> Rect a -> Rect a -> Rect a
instance GHC.Generics.Generic (NumHask.Data.Rect.Rect a)
instance Data.Traversable.Traversable NumHask.Data.Rect.Rect
instance Data.Foldable.Foldable NumHask.Data.Rect.Rect
instance GHC.Base.Applicative NumHask.Data.Rect.Rect
instance GHC.Base.Functor NumHask.Data.Rect.Rect
instance GHC.Classes.Eq a => GHC.Classes.Eq (NumHask.Data.Rect.Rect a)
instance GHC.Show.Show a => GHC.Show.Show (NumHask.Data.Rect.Rect a)
instance Data.Distributive.Distributive NumHask.Data.Rect.Rect
instance Data.Functor.Rep.Representable NumHask.Data.Rect.Rect
instance NumHask.Algebra.Abstract.Lattice.BoundedLattice a => GHC.Base.Semigroup (NumHask.Data.Rect.Rect a)
instance NumHask.Algebra.Abstract.Lattice.BoundedLattice a => GHC.Base.Monoid (NumHask.Data.Rect.Rect a)
instance NumHask.Algebra.Abstract.Lattice.Lattice a => NumHask.Analysis.Space.Space (NumHask.Data.Rect.Rect a)
instance (NumHask.Algebra.Abstract.Lattice.Lattice a, NumHask.Algebra.Abstract.Field.Field a, NumHask.Algebra.Abstract.Additive.Subtractive a, NumHask.Data.Integral.FromInteger a) => NumHask.Analysis.Space.FieldSpace (NumHask.Data.Rect.Rect a)
instance NumHask.Algebra.Abstract.Additive.Additive a => NumHask.Algebra.Abstract.Action.AdditiveAction (NumHask.Data.Rect.Rect a)
instance NumHask.Algebra.Abstract.Additive.Subtractive a => NumHask.Algebra.Abstract.Action.SubtractiveAction (NumHask.Data.Rect.Rect a)
instance NumHask.Algebra.Abstract.Multiplicative.Multiplicative a => NumHask.Algebra.Abstract.Action.MultiplicativeAction (NumHask.Data.Rect.Rect a)
instance NumHask.Algebra.Abstract.Multiplicative.Divisive a => NumHask.Algebra.Abstract.Action.DivisiveAction (NumHask.Data.Rect.Rect a)