# Library Coq.ZArith.Zabs

``` ```
 Binary Integers (Pierre Crégut (CNET, Lannion, France)
``` Require Import Arith. Require Import BinPos. Require Import BinInt. Require Import Zorder. Require Import ZArith_dec. Open Local Scope Z_scope. ```
 Properties of absolute value
``` Lemma Zabs_eq : forall n:Z, 0 <= n -> Zabs n = n. intro x; destruct x; auto with arith. compute in |- *; intros; absurd (Gt = Gt); trivial with arith. Qed. Lemma Zabs_non_eq : forall n:Z, n <= 0 -> Zabs n = - n. Proof. intro x; destruct x; auto with arith. compute in |- *; intros; absurd (Gt = Gt); trivial with arith. Qed. Theorem Zabs_Zopp : forall n:Z, Zabs (- n) = Zabs n. Proof. intros z; case z; simpl in |- *; auto. Qed. ```
 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 (Zabs n). Proof. intros P x H H0; elim (Z_lt_ge_dec x 0); intro. assert (x <= 0). apply Zlt_le_weak; assumption. rewrite Zabs_non_eq. apply H0. assumption. assumption. rewrite Zabs_eq. apply H; assumption. apply Zge_le. assumption. Qed. Theorem Zabs_intro : forall P (n:Z), P (- n) -> P n -> P (Zabs n). intros P z; case z; simpl in |- *; auto. Qed. Definition Zabs_dec : forall x:Z, {x = Zabs x} + {x = - Zabs x}. Proof. intro x; destruct x; auto with arith. Defined. Lemma Zabs_pos : forall n:Z, 0 <= Zabs n. intro x; destruct x; auto with arith; compute in |- *; intros H; inversion H. Qed. Theorem Zabs_eq_case : forall n m:Z, Zabs n = Zabs m -> n = m \/ n = - m. Proof. intros z1 z2; case z1; case z2; simpl in |- *; auto;  try (intros; discriminate); intros p1 p2 H1; injection H1;  (intros H2; rewrite H2); auto. Qed. ```
 Triangular inequality
``` Hint Local Resolve Zle_neg_pos: zarith. Theorem Zabs_triangle : forall n m:Z, Zabs (n + m) <= Zabs n + Zabs m. Proof. intros z1 z2; case z1; case z2; try (simpl in |- *; auto with zarith; fail). intros p1 p2;  apply Zabs_intro with (P:= fun x => x <= Zabs (Zpos p2) + Zabs (Zneg p1));  try rewrite Zopp_plus_distr; auto with zarith. apply Zplus_le_compat; simpl in |- *; auto with zarith. apply Zplus_le_compat; simpl in |- *; auto with zarith. intros p1 p2;  apply Zabs_intro with (P:= fun x => x <= Zabs (Zpos p2) + Zabs (Zneg p1));  try rewrite Zopp_plus_distr; auto with zarith. apply Zplus_le_compat; simpl in |- *; auto with zarith. apply Zplus_le_compat; simpl in |- *; auto with zarith. Qed. ```
 Absolute value and multiplication
``` Lemma Zsgn_Zabs : forall n:Z, n * Zsgn n = Zabs n. Proof. intro x; destruct x; rewrite Zmult_comm; auto with arith. Qed. Lemma Zabs_Zsgn : forall n:Z, Zabs n * Zsgn n = n. Proof. intro x; destruct x; rewrite Zmult_comm; auto with arith. Qed. Theorem Zabs_Zmult : forall n m:Z, Zabs (n * m) = Zabs n * Zabs m. Proof. intros z1 z2; case z1; case z2; simpl in |- *; auto. Qed. ```
 absolute value in nat is compatible with order
``` Lemma Zabs_nat_lt :  forall n m:Z, 0 <= n /\ n < m -> (Zabs_nat n < Zabs_nat m)%nat. Proof. intros x y. case x; simpl in |- *. case y; simpl in |- *. intro. absurd (0 < 0). compute in |- *. intro H0. discriminate H0. intuition. intros. elim (ZL4 p). intros. rewrite H0. auto with arith. intros. elim (ZL4 p). intros. rewrite H0. auto with arith. case y; simpl in |- *. intros. absurd (Zpos p < 0). compute in |- *. intro H0. discriminate H0. intuition. intros. change (nat_of_P p > nat_of_P p0)%nat in |- *. apply nat_of_P_gt_Gt_compare_morphism. elim H; auto with arith. intro. exact (ZC2 p0 p). intros. absurd (Zpos p0 < Zneg p). compute in |- *. intro H0. discriminate H0. intuition. intros. absurd (0 <= Zneg p). compute in |- *. auto with arith. intuition. Qed.```
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