(** The corresponding order is a strict order. **)
Definition comparableLt {A:Type} (C: Comparable A) : relation A :=
  fun x y => compare x y = Lt.Instance ComparableLtStrictOrder {A:Type} (C: Comparable A) :
  StrictOrder (comparableLt C).(** nat is comparable. **)
Program Instance natComparable : Comparable nat :=
  { compare := Nat.compare }.(** A pair of comparable is comparable. **)
Program Instance PairComparable {A:Type} (CA:Comparable A) {B:Type} (CB:Comparable B) :
  Comparable (A*B) :=
  { compare := fun x y =>
      let (xa, xb) := x in let (ya, yb) := y in
      match compare xa ya return comparison with
        | Eq => compare xb yb
        | x => x
      end }.(** Special case of comparable, where equality is usual equality. **)
Class ComparableUsualEq {A:Type} (C: Comparable A) :=
  compare_eq : forall x y, compare x y = Eq -> x = y.(** Special case of comparable, where equality is usual equality. **)
Class ComparableUsualEq {A:Type} (C: Comparable A) :=
  compare_eq : forall x y, compare x y = Eq -> x = y.(** Boolean equality for a [Comparable]. **)
Definition compare_eqb {A:Type} {C:Comparable A} (x y:A) :=
  match compare x y with
    | Eq => true
    | _ => false
  end.

Theorem compare_eqb_iff {A:Type} {C:Comparable A} {U:ComparableUsualEq C} : forall x y, compare_eqb x y = true <-> x = y.