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(*         *   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)           *)
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(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
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(** Binary Integers : properties of absolute value *)
(** Initial author : Pierre Crégut (CNET, Lannion, France) *)
(** THIS FILE IS DEPRECATED. It is now almost entirely made of compatibility formulations for results already present in BinInt.Z. *)
Local Open Scope Z_scope.
(**********************************************************************)
(** * Properties of absolute value *)
Notation Zabs_non_eq := Z.abs_neq (only parsing).
Notation Zabs_Zopp := Z.abs_opp (only parsing).
Notation Zabs_pos := Z.abs_nonneg (only parsing).
Notation Zabs_eq_case := Z.abs_eq_cases (only parsing).
Notation Zsgn_Zabs := Z.sgn_abs (only parsing).
Notation Zabs_Zsgn := Z.abs_sgn (only parsing).
Notation Zabs_Zmult := Z.abs_mul (only parsing).
(** * Proving a property of the absolute value by cases *)
Definition Zabs_dec : forall x:Z, {x = Z.abs x} + {x = - Z.abs x}.
(** * Some results about the sign function. *)
Notation Zsgn_Zmult := Z.sgn_mul (only parsing).
Notation Zsgn_Zopp := Z.sgn_opp (only parsing).
Notation Zsgn_pos := Z.sgn_pos_iff (only parsing).
Notation Zsgn_neg := Z.sgn_neg_iff (only parsing).
Notation Zsgn_null := Z.sgn_null_iff (only parsing).
(** A characterization of the sign function: *)
(** Compatibility *)
Notation inj_Zabs_nat := Zabs2Nat.id_abs (only parsing).
Notation Zabs_nat_Z_of_nat := Zabs2Nat.id (only parsing).
Notation Zabs_nat_mult := Zabs2Nat.inj_mul (only parsing).
Notation Zabs_nat_Zsucc := Zabs2Nat.inj_succ (only parsing).
Notation Zabs_nat_Zplus := Zabs2Nat.inj_add (only parsing).
Notation Zabs_nat_Zminus := (fun n m => Zabs2Nat.inj_sub m n) (only parsing).
Notation Zabs_nat_compare := Zabs2Nat.inj_compare (only parsing).

Lemma Zabs_nat_le n m : 0 <= n <= m -> (Z.abs_nat n <= Z.abs_nat m)%nat.