Library Stdlib.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.

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

#[local] Open Scope Z_scope.

Properties of absolute value

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.

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

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.