Definition Req_bool x y :=
  match Rcompare x y with
  | Eq => true
  | _ => false
  end.Inductive Req_bool_prop (x y : R) : bool -> Prop :=
  | Req_bool_true_ : (x = y)%R -> Req_bool_prop x y true
  | Req_bool_false_ : (x <> y)%R -> Req_bool_prop x y false.Inductive Req_bool_prop (x y : R) : bool -> Prop :=
  | Req_bool_true_ : (x = y)%R -> Req_bool_prop x y true
  | Req_bool_false_ : (x <> y)%R -> Req_bool_prop x y false.Inductive Req_bool_prop (x y : R) : bool -> Prop :=
  | Req_bool_true_ : (x = y)%R -> Req_bool_prop x y true
  | Req_bool_false_ : (x <> y)%R -> Req_bool_prop x y false.

Theorem Req_bool_false : forall x y, (x <> y)%R -> Req_bool x y = false.