19 Data.Ix
Date: 2010-07-20 13:11:00

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## Chapter 19
Data.Ix

module Data.Ix (
Ix(range, index, inRange, rangeSize)
) where

### 19.1 The Ix class

class Ord a => Ix a where

  The Ix class is used to map a contiguous subrange of values in a type onto integers. It is used primarily for array indexing (see the array package).

  The first argument (l,u) of each of these operations is a pair specifying the lower and upper bounds of a contiguous subrange of values.

  An implementation is entitled to assume the following laws about these operations:

  - inRange (l,u) i == elem i (range (l,u))
  - range (l,u) !! index (l,u) i == i, when inRange (l,u) i
  - map (index (l,u)) (range (l,u))) == [0..rangeSize (l,u)-1]
  - rangeSize (l,u) == length (range (l,u))

  Minimal complete instance: range, index and inRange.

  Methods

  range :: (a, a) -> [a]

    The list of values in the subrange defined by a bounding pair.

  index :: (a, a) -> a -> Int

    The position of a subscript in the subrange.

  inRange :: (a, a) -> a -> Bool

    Returns True the given subscript lies in the range defined the bounding pair.

  rangeSize :: (a, a) -> Int

    The size of the subrange defined by a bounding pair.

instance Ix Bool
instance Ix Char
instance Ix Int
instance Ix Int8
instance Ix Int16
instance Ix Int32
instance Ix Int64
instance Ix Integer
instance Ix Ordering
instance Ix Word
instance Ix Word8
instance Ix Word16
instance Ix Word32
instance Ix Word64
instance Ix ()
instance Ix GeneralCategory
instance Ix SeekMode
instance Ix IOMode
instance (Ix a, Ix b) => Ix (a, b)
instance (Ix a1, Ix a2, Ix a3) => Ix (a1, a2, a3)
instance (Ix a1, Ix a2, Ix a3, Ix a4) => Ix (a1, a2, a3, a4)
instance (Ix a1, Ix a2, Ix a3, Ix a4, Ix a5) => Ix (a1, a2, a3, a4, a5)

### 19.2 Deriving Instances of Ix

It is possible to derive an instance of Ix automatically, using a deriving clause on a data declaration. Such derived instance declarations for the class Ix are only possible for enumerations (i.e. datatypes having only nullary constructors) and single-constructor datatypes, whose constituent types are instances of Ix. A Haskell implementation must provide Ix instances for tuples up to at least size 15.

For an enumeration, the nullary constructors are assumed to be numbered left-to-right with the indices being 0 to n-1 inclusive. This is the same numbering defined by the Enum class. For example, given the datatype:

data Colour = Red | Orange | Yellow | Green | Blue | Indigo | Violet

we would have:

range (Yellow,Blue) == [Yellow,Green,Blue]
index (Yellow,Blue) Green == 1
inRange (Yellow,Blue) Red == False

For single-constructor datatypes, the derived instance declarations are as shown for tuples:

instance (Ix a, Ix b) => Ix (a,b) where
range ((l,l'),(u,u'))
= [(i,i') | i <- range (l,u), i' <- range (l',u')]
index ((l,l'),(u,u')) (i,i')
= index (l,u) i ⋆ rangeSize (l',u') + index (l',u') i'
inRange ((l,l'),(u,u')) (i,i')
= inRange (l,u) i && inRange (l',u') i'
-- Instances for other tuples are obtained from this scheme:
--
-- instance (Ix a1, Ix a2, ... , Ix ak) => Ix (a1,a2,...,ak) where
-- range ((l1,l2,...,lk),(u1,u2,...,uk)) =
-- [(i1,i2,...,ik) | i1 <- range (l1,u1),
-- i2 <- range (l2,u2),
-- ...
-- ik <- range (lk,uk)]
--
-- index ((l1,l2,...,lk),(u1,u2,...,uk)) (i1,i2,...,ik) =
-- index (lk,uk) ik + rangeSize (lk,uk) ⋆ (
-- index (lk-1,uk-1) ik-1 + rangeSize (lk-1,uk-1) ⋆ (
-- ...
-- index (l1,u1)))
--
-- inRange ((l1,l2,...lk),(u1,u2,...,uk)) (i1,i2,...,ik) =
-- inRange (l1,u1) i1 && inRange (l2,u2) i2 &&
-- ... && inRange (lk,uk) ik

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