(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)
(**********************************************************)
(** Complements for the reals.Integer and fractional part *)
(*                                                        *)
(**********************************************************)
Local Open Scope R_scope.
(*********************************************************)
(** *    Fractional part                                 *)
(*********************************************************)
(**********)
Definition Int_part (r:R) : Z := (up r - 1)%Z.
(**********)
Definition frac_part (r:R) : R := r - IZR (Int_part r).
(*********************************************************)
(** *    Properties                                      *)
(*********************************************************)

(**********)
Lemma base_Int_part :
forall r:R, IZR (Int_part r) <= r /\ IZR (Int_part r) - r > -1.