(** A comparable type is equiped with a [compare] function, that define an order
   relation. **)
Class Comparable (A:Type) := {
  compare : A -> A -> comparison;
  compare_antisym : forall x y, CompOpp (compare x y) = compare y x;
  compare_trans :  forall x y z c,
    (compare x y) = c -> (compare y z) = c -> (compare x z) = c
}.(** A comparable type is equiped with a [compare] function, that define an order
   relation. **)
Class Comparable (A:Type) := {
  compare : A -> A -> comparison;
  compare_antisym : forall x y, CompOpp (compare x y) = compare y x;
  compare_trans :  forall x y z c,
    (compare x y) = c -> (compare y z) = c -> (compare x z) = c
}.(** A comparable type is equiped with a [compare] function, that define an order
   relation. **)
Class Comparable (A:Type) := {
  compare : A -> A -> comparison;
  compare_antisym : forall x y, CompOpp (compare x y) = compare y x;
  compare_trans :  forall x y z c,
    (compare x y) = c -> (compare y z) = c -> (compare x z) = c
}.(** A comparable type is equiped with a [compare] function, that define an order
   relation. **)
Class Comparable (A:Type) := {
  compare : A -> A -> comparison;
  compare_antisym : forall x y, CompOpp (compare x y) = compare y x;
  compare_trans :  forall x y z c,
    (compare x y) = c -> (compare y z) = c -> (compare x z) = c
}.

Theorem compare_refl {A:Type} (C: Comparable A) : forall x, compare x x = Eq.