Definition Rle_bool x y :=
  match Rcompare x y with
  | Gt => false
  | _ => true
  end.Inductive Rle_bool_prop (x y : R) : bool -> Prop :=
  | Rle_bool_true_ : (x <= y)%R -> Rle_bool_prop x y true
  | Rle_bool_false_ : (y < x)%R -> Rle_bool_prop x y false.Inductive Rle_bool_prop (x y : R) : bool -> Prop :=
  | Rle_bool_true_ : (x <= y)%R -> Rle_bool_prop x y true
  | Rle_bool_false_ : (y < x)%R -> Rle_bool_prop x y false.Inductive Rle_bool_prop (x y : R) : bool -> Prop :=
  | Rle_bool_true_ : (x <= y)%R -> Rle_bool_prop x y true
  | Rle_bool_false_ : (y < x)%R -> Rle_bool_prop x y false.

(* Specifications: *)
Theorem Rle_bool_spec : forall x y, Rle_bool_prop x y (Rle_bool x y).
Theorem Rle_bool_true : forall x y, (x <= y)%R -> Rle_bool x y = true.
Theorem Rle_bool_false : forall x y, (y < x)%R -> Rle_bool x y = false.
Theorem Rle_bool_spec : forall x y, Rle_bool_prop x y (Rle_bool x y).
Theorem Rle_bool_true : forall x y, (x <= y)%R -> Rle_bool x y = true.
Theorem Rle_bool_false : forall x y, (y < x)%R -> Rle_bool x y = false.