```module Stdlib.Trait.Ord;

import Stdlib.Data.Fixity open;
import Stdlib.Data.Bool.Base open;
import Stdlib.Trait.Eq open;

type Ordering :=
| LT : Ordering
| EQ : Ordering
| GT : Ordering;

isLT : Ordering → Bool
| LT := true
| _ := false;

isEQ : Ordering → Bool
| EQ := true
| _ := false;

isGT : Ordering → Bool
| GT := true
| _ := false;

instance
orderingEqI : Eq Ordering :=
mkEq@{
eq (x : Ordering) : Ordering -> Bool
| LT := isLT x
| EQ := isEQ x
| GT := isGT x
};

--- A trait for defining a total order
trait
type Ord A := mkOrd {cmp : A -> A -> Ordering};

syntax operator <= comparison;

--- Returns ;true; iff the first element is less than or equal to the second.
{-# inline: always #-}
<= {A} {{Ord A}} (x y : A) : Bool :=
case Ord.cmp x y of {
| EQ := true
| LT := true
| GT := false
};

syntax operator < comparison;

--- Returns ;true; iff the first element is less than the second.
{-# inline: always #-}
< {A} {{Ord A}} (x y : A) : Bool :=
case Ord.cmp x y of {
| EQ := false
| LT := true
| GT := false
};

syntax operator > comparison;

--- Returns ;true; iff the first element is greater than the second.
{-# inline: always #-}
> {A} {{Ord A}} (x y : A) : Bool := y < x;

syntax operator >= comparison;

--- Returns ;true; iff the first element is greater than or equal to the second.
{-# inline: always #-}
>= {A} {{Ord A}} (x y : A) : Bool := y <= x;

--- Returns the smaller element.
{-# inline: always #-}
min {A} {{Ord A}} (x y : A) : A := if (x < y) x y;

--- Returns the larger element.
{-# inline: always #-}
max {A} {{Ord A}} (x y : A) : A := if (x > y) x y;
```