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(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \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)         *)
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(** * An absolute value for normalized rational numbers. *)
(** Contributed by Cédric Auger *)

(* Specifications: *)
Lemma Qred_abs (x : Q) : Qred (Qabs x) = Qabs (Qred x).
Lemma Qcabs_canon (x : Q) : Qred x = x -> Qred (Qabs x) = Qabs x.
Lemma Qcabs_pos x : 0 <= x -> [x] = x.
Lemma Qcabs_neg x : x <= 0 -> [x] = - x.
Lemma Qcabs_nonneg x : 0 <= [x].
Lemma Qcabs_triangle x y : [x+y] <= [x] + [y].
Lemma Qcabs_Qcmult x y : [x*y] = [x]*[y].
Lemma Qcabs_Qcminus x y: [x-y] = [y-x].
Lemma Qcle_Qcabs x : x <= [x].
Lemma Qcabs_Qcle_condition x y : [x] <= y <-> -y <= x <= y.
Lemma Qcabs_diff_Qcle_condition x y r : [x-y] <= r <-> x - r <= y <= x + r.
Lemma Qcabs_null x : [x] = 0 -> x = 0.