# Library Coq.ZArith.Zabs

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.

Require Import Arith_base.
Require Import BinPos.
Require Import BinInt.
Require Import Zcompare.
Require Import Zorder.
Require Import Znat.
Require Import ZArith_dec.

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

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}.

Lemma Zabs_spec x :
0 <= x /\ Z.abs x = x \/
0 > x /\ Z.abs x = -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:

Lemma Zsgn_spec x :
0 < x /\ Z.sgn x = 1 \/
0 = x /\ Z.sgn x = 0 \/
0 > x /\ Z.sgn x = -1.

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.

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