## Concept:

# Type class Ord

## Headline

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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