(* -*- coding: utf-8 -*- *)
(************************************************************************)
(*         *      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)         *)
(************************************************************************)
(** 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 *)

(* Specifications: *)
Lemma Zabs_ind :
forall (P:Z -> Prop) (n:Z),
(n >= 0 -> P n) -> (n <= 0 -> P (- n)) -> P (Z.abs n).
Theorem Zabs_intro : forall P (n:Z), P (- n) -> P n -> P (Z.abs n).
Definition Zabs_dec : forall x:Z, {x = Z.abs x} + {x = - Z.abs x}.