Definition Rlt_bool x y :=
  match Rcompare x y with
  | Lt => true
  | _ => false
  end.Inductive Rlt_bool_prop (x y : R) : bool -> Prop :=
  | Rlt_bool_true_ : (x < y)%R -> Rlt_bool_prop x y true
  | Rlt_bool_false_ : (y <= x)%R -> Rlt_bool_prop x y false.Inductive Rlt_bool_prop (x y : R) : bool -> Prop :=
  | Rlt_bool_true_ : (x < y)%R -> Rlt_bool_prop x y true
  | Rlt_bool_false_ : (y <= x)%R -> Rlt_bool_prop x y false.Inductive Rlt_bool_prop (x y : R) : bool -> Prop :=
  | Rlt_bool_true_ : (x < y)%R -> Rlt_bool_prop x y true
  | Rlt_bool_false_ : (y <= x)%R -> Rlt_bool_prop x y false.

(* Specifications: *)
Theorem Rlt_bool_spec : forall x y, Rlt_bool_prop x y (Rlt_bool x y).
Theorem negb_Rlt_bool : forall x y, negb (Rle_bool x y) = Rlt_bool y x.
Theorem negb_Rle_bool : forall x y, negb (Rlt_bool x y) = Rle_bool y x.
Theorem Rlt_bool_true : forall x y, (x < y)%R -> Rlt_bool x y = true.
Theorem Rlt_bool_false : forall x y, (y <= x)%R -> Rlt_bool x y = false.
Theorem Rlt_bool_spec : forall x y, Rlt_bool_prop x y (Rlt_bool x y).
Theorem negb_Rlt_bool : forall x y, negb (Rle_bool x y) = Rlt_bool y x.
Theorem negb_Rle_bool : forall x y, negb (Rlt_bool x y) = Rle_bool y x.
Theorem Rlt_bool_true : forall x y, (x < y)%R -> Rlt_bool x y = true.
Theorem Rlt_bool_false : forall x y, (y <= x)%R -> Rlt_bool x y = false.