```module Stdlib.Data.Nat.Ord;

import Stdlib.Data.Fixity open;

import Stdlib.Data.Nat.Base open;
import Stdlib.Trait.Ord open using {EQ; LT; GT; Ordering};
import Stdlib.Data.Bool.Base open;

syntax operator == comparison;

--- Tests for equality.
builtin nat-eq
== : Nat → Nat → Bool
| zero zero := true
| zero _ := false
| _ zero := false
| (suc n) (suc m) := n == m;

syntax operator /= comparison;

--- Tests for inequality.
/= (x y : Nat) : Bool := not (x == y);

syntax operator <= comparison;

--- Returns ;true; iff the first element is less than or equal to the second.
builtin nat-le
<= : Nat → Nat → Bool
| zero _ := true
| _ zero := false
| (suc n) (suc m) := n <= m;

syntax operator < comparison;

--- Returns ;true; iff the first element is less than the second.
builtin nat-lt
< (n m : Nat) : Bool := suc n <= m;

syntax operator > comparison;

--- Returns ;true; iff the first element is greater than the second.
> (n m : Nat) : Bool := m < n;

syntax operator >= comparison;

--- Returns ;true; iff the first element is greater than or equal to the second.
>= (n m : Nat) : Bool := m <= n;

--- Tests for ;Ordering;.
{-# inline: true #-}
compare (n m : Nat) : Ordering := if (n == m) EQ (if (n < m) LT GT);

--- Returns the smaller ;Nat;.
min (x y : Nat) : Nat := if (x < y) x y;

--- Returns the larger ;Nat;.
max (x y : Nat) : Nat := if (x > y) x y;
```