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A Language:Haskell type class for ordering

Description

The type class is concerned with types that admit comparison.

class Eq a => Ord a where
  compare :: a -> a -> Ordering
  (<) :: a -> a -> Bool
  (<=) :: a -> a -> Bool
  (>) :: a -> a -> Bool
  (>=) :: a -> a -> Bool
  max :: a -> a -> a
  min :: a -> a -> a

These are essentially familiar comparison operators.

compare returns a value of an enum type to describing ordering.

data Ordering = LT | EQ | GT

LT should be read as "less than", EQ as "equal", and GT as "greater than".

It is sufficient to either define compare or "<=". All the other operators can be derived in a regular manner. Here is important to notice that every instance of Ord must also be an instance of Eq; see the type-class constraint above. Thus, < ad <= can be easily distinguished with the help of Eq.

Illustration

Consider the following data type for natural numbers: the Zero constructor should be thought as representing "0"; the Succ constructor should be thought of as representing the successor function (i.e., addition of 1 or increment).

data Nat = Zero | Succ Nat

Equality and ordering could be reasonably defined for Nat as follows.

instance Eq Nat where
  Zero == Zero = True
  Zero == (Succ _) = False
  (Succ _) == Zero = False
  (Succ x) == (Succ y) = x == y

instance Ord Nat where
  compare Zero Zero = EQ
  compare Zero (Succ _) = LT
  compare (Succ _) Zero = GT
  compare (Succ x) (Succ y) = compare x y

Equality and ordering is exercised at the Haskell prompt as follows.

> Zero == Zero
True
> Zero == Succ Zero
False
> Succ Zero == Succ Zero
True
> Zero < Succ Zero
True
> Succ Zero < Succ Zero
False


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