15  Indexing Operations

module Ix ( Ix(range, index, inRange, rangeSize) ) where class  Ord a => Ix a  where     range       :: (a,a) -> [a]     index       :: (a,a) -> a -> Int     inRange     :: (a,a) -> a -> Bool     rangeSize   :: (a,a) -> Int instance                   Ix Char      where ... instance                   Ix Int       where ... instance                   Ix Integer   where ... instance  (Ix a, Ix b)  => Ix (a,b)     where ... -- et cetera instance                   Ix Bool      where ... instance                   Ix Ordering  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 Chapter 16). The Ix class contains the methods range, index, and inRange. The index operation maps a bounding pair, which defines the lower and upper bounds of the range, and a subscript, to an integer. The range operation enumerates all subscripts; the inRange operation tells whether a particular subscript lies in the range defined by a bounding pair.

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

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

15.1  Deriving Instances of Ix

It is possible to derive an instance of Ix automatically, using a deriving clause on a data declaration (Section 4.3.3). 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 in Figure 15.1.

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 Derivation of Ix instances

15.2  Library Ix