# Library Coq.ZArith.auxiliary

``` ```
Binary Integers (Pierre Crégut, CNET, Lannion, France)
``` Require Export Arith_base. Require Import BinInt. Require Import Zorder. Require Import Decidable. Require Import Peano_dec. Require Export Compare_dec. Open Local Scope Z_scope. ```

# Moving terms from one side to the other of an inequality

``` Theorem Zne_left : forall n m:Z, Zne n m -> Zne (n + - m) 0. Proof.   intros x y; unfold Zne in |- *; unfold not in |- *; intros H1 H2; apply H1;     apply Zplus_reg_l with (- y); rewrite Zplus_opp_l;       rewrite Zplus_comm; trivial with arith. Qed. Theorem Zegal_left : forall n m:Z, n = m -> n + - m = 0. Proof.   intros x y H; apply (Zplus_reg_l y); rewrite Zplus_permute;     rewrite Zplus_opp_r; do 2 rewrite Zplus_0_r; assumption. Qed. Theorem Zle_left : forall n m:Z, n <= m -> 0 <= m + - n. Proof.   intros x y H; replace 0 with (x + - x).   apply Zplus_le_compat_r; trivial.   apply Zplus_opp_r. Qed. Theorem Zle_left_rev : forall n m:Z, 0 <= m + - n -> n <= m. Proof.   intros x y H; apply Zplus_le_reg_r with (- x).   rewrite Zplus_opp_r; trivial. Qed. Theorem Zlt_left_rev : forall n m:Z, 0 < m + - n -> n < m. Proof.   intros x y H; apply Zplus_lt_reg_r with (- x).   rewrite Zplus_opp_r; trivial. Qed. Theorem Zlt_left : forall n m:Z, n < m -> 0 <= m + -1 + - n. Proof.   intros x y H; apply Zle_left; apply Zsucc_le_reg;     change (Zsucc x <= Zsucc (Zpred y)) in |- *; rewrite <- Zsucc_pred;       apply Zlt_le_succ; assumption. Qed. Theorem Zlt_left_lt : forall n m:Z, n < m -> 0 < m + - n. Proof.   intros x y H; replace 0 with (x + - x).   apply Zplus_lt_compat_r; trivial.   apply Zplus_opp_r. Qed. Theorem Zge_left : forall n m:Z, n >= m -> 0 <= n + - m. Proof.   intros x y H; apply Zle_left; apply Zge_le; assumption. Qed. Theorem Zgt_left : forall n m:Z, n > m -> 0 <= n + -1 + - m. Proof.   intros x y H; apply Zlt_left; apply Zgt_lt; assumption. Qed. Theorem Zgt_left_gt : forall n m:Z, n > m -> n + - m > 0. Proof.   intros x y H; replace 0 with (y + - y).   apply Zplus_gt_compat_r; trivial.   apply Zplus_opp_r. Qed. Theorem Zgt_left_rev : forall n m:Z, n + - m > 0 -> n > m. Proof.   intros x y H; apply Zplus_gt_reg_r with (- y).   rewrite Zplus_opp_r; trivial. Qed. ```

# Factorization lemmas

``` Theorem Zred_factor0 : forall n:Z, n = n * 1.   intro x; rewrite (Zmult_1_r x); reflexivity. Qed. Theorem Zred_factor1 : forall n:Z, n + n = n * 2. Proof.   exact Zplus_diag_eq_mult_2. Qed. Theorem Zred_factor2 : forall n m:Z, n + n * m = n * (1 + m). Proof.   intros x y; pattern x at 1 in |- *; rewrite <- (Zmult_1_r x);     rewrite <- Zmult_plus_distr_r; trivial with arith. Qed. Theorem Zred_factor3 : forall n m:Z, n * m + n = n * (1 + m). Proof.   intros x y; pattern x at 2 in |- *; rewrite <- (Zmult_1_r x);     rewrite <- Zmult_plus_distr_r; rewrite Zplus_comm;       trivial with arith. Qed. Theorem Zred_factor4 : forall n m p:Z, n * m + n * p = n * (m + p). Proof.   intros x y z; symmetry in |- *; apply Zmult_plus_distr_r. Qed. Theorem Zred_factor5 : forall n m:Z, n * 0 + m = m. Proof.   intros x y; rewrite <- Zmult_0_r_reverse; auto with arith. Qed. Theorem Zred_factor6 : forall n:Z, n = n + 0. Proof.   intro; rewrite Zplus_0_r; trivial with arith. Qed. Theorem Zle_mult_approx :   forall n m p:Z, n > 0 -> p > 0 -> 0 <= m -> 0 <= m * n + p. Proof.   intros x y z H1 H2 H3; apply Zle_trans with (m := y * x);     [ apply Zmult_gt_0_le_0_compat; assumption | pattern (y * x) at 1 in |- *; rewrite <- Zplus_0_r; apply Zplus_le_compat_l; apply Zlt_le_weak; apply Zgt_lt; assumption ]. Qed. Theorem Zmult_le_approx :   forall n m p:Z, n > 0 -> n > p -> 0 <= m * n + p -> 0 <= m. Proof.   intros x y z H1 H2 H3; apply Zlt_succ_le; apply Zmult_gt_0_lt_0_reg_r with x;     [ assumption       | apply Zle_lt_trans with (1 := H3); rewrite <- Zmult_succ_l_reverse;         apply Zplus_lt_compat_l; apply Zgt_lt; assumption ]. Qed. ```