The Haskell 98 Report
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## 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`

`
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
rangeSize b@(l,h) | null (range b) = 0
| otherwise = index b h + 1
-- NB: replacing "null (range b)" by "not (l <= h)"
-- fails if the bounds are tuples. For example,
-- (1,2) <= (2,1)
-- but the range is nevertheless empty
-- range ((1,2),(2,1)) = []
instance Ix Char where
range (m,n) = [m..n]
index b@(c,c') ci
| inRange b ci = fromEnum ci - fromEnum c
| otherwise = error "Ix.index: Index out of range."
inRange (c,c') i = c <= i && i <= c'
instance Ix Int where
range (m,n) = [m..n]
index b@(m,n) i
| inRange b i = i - m
| otherwise = error "Ix.index: Index out of range."
inRange (m,n) i = m <= i && i <= n
instance Ix Integer where
range (m,n) = [m..n]
index b@(m,n) i
| inRange b i = fromInteger (i - m)
| otherwise = error "Ix.index: Index out of range."
inRange (m,n) i = m <= i && i <= n
instance (Ix a,Ix b) => Ix (a, b) -- as derived, for all tuples
instance Ix Bool -- as derived
instance Ix Ordering -- as derived
instance Ix () -- as derived

The Haskell 98 Report
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December 2002`