Library Coq.ZArith.Zorder

``` ```
Binary Integers (Pierre Crégut (CNET, Lannion, France)
``` Require Import BinPos. Require Import BinInt. Require Import Arith_base. Require Import Decidable. Require Import Zcompare. Open Local Scope Z_scope. Implicit Types x y z : Z. ```
Properties of the order relations on binary integers
``` ```

Trichotomy

``` Theorem Ztrichotomy_inf : forall n m:Z, {n < m} + {n = m} + {n > m}. Proof.   unfold Zgt, Zlt in |- *; intros m n; assert (H := refl_equal (m ?= n)).   set (x := m ?= n) in H at 2 |- *.   destruct x;     [ left; right; rewrite Zcompare_Eq_eq with (1 := H) | left; left | right ];     reflexivity. Qed. Theorem Ztrichotomy : forall n m:Z, n < m \/ n = m \/ n > m. Proof.   intros m n; destruct (Ztrichotomy_inf m n) as [[Hlt| Heq]| Hgt];     [ left | right; left | right; right ]; assumption. Qed. ```

Decidability of equality and order on Z

``` Theorem dec_eq : forall n m:Z, decidable (n = m). Proof.   intros x y; unfold decidable in |- *; elim (Zcompare_Eq_iff_eq x y);     intros H1 H2; elim (Dcompare (x ?= y));       [ tauto | intros H3; right; unfold not in |- *; intros H4; elim H3; rewrite (H2 H4); intros H5; discriminate H5 ]. Qed. Theorem dec_Zne : forall n m:Z, decidable (Zne n m). Proof.   intros x y; unfold decidable, Zne in |- *; elim (Zcompare_Eq_iff_eq x y).   intros H1 H2; elim (Dcompare (x ?= y));     [ right; rewrite H1; auto       | left; unfold not in |- *; intro; absurd ((x ?= y) = Eq);         [ elim H; intros HR; rewrite HR; discriminate | auto ] ]. Qed. Theorem dec_Zle : forall n m:Z, decidable (n <= m). Proof.   intros x y; unfold decidable, Zle in |- *; elim (x ?= y);     [ left; discriminate | left; discriminate | right; unfold not in |- *; intros H; apply H; trivial with arith ]. Qed. Theorem dec_Zgt : forall n m:Z, decidable (n > m). Proof.   intros x y; unfold decidable, Zgt in |- *; elim (x ?= y);     [ right; discriminate | right; discriminate | auto with arith ]. Qed. Theorem dec_Zge : forall n m:Z, decidable (n >= m). Proof.   intros x y; unfold decidable, Zge in |- *; elim (x ?= y);     [ left; discriminate | right; unfold not in |- *; intros H; apply H; trivial with arith | left; discriminate ]. Qed. Theorem dec_Zlt : forall n m:Z, decidable (n < m). Proof.   intros x y; unfold decidable, Zlt in |- *; elim (x ?= y);     [ right; discriminate | auto with arith | right; discriminate ]. Qed. Theorem not_Zeq : forall n m:Z, n <> m -> n < m \/ m < n. Proof.   intros x y; elim (Dcompare (x ?= y));     [ intros H1 H2; absurd (x = y);       [ assumption | elim (Zcompare_Eq_iff_eq x y); auto with arith ]       | unfold Zlt in |- *; intros H; elim H; intros H1;         [ auto with arith | right; elim (Zcompare_Gt_Lt_antisym x y); auto with arith ] ]. Qed. ```

Relating strict and large orders

``` Lemma Zgt_lt : forall n m:Z, n > m -> m < n. Proof.   unfold Zgt, Zlt in |- *; intros m n H; elim (Zcompare_Gt_Lt_antisym m n);     auto with arith. Qed. Lemma Zlt_gt : forall n m:Z, n < m -> m > n. Proof.   unfold Zgt, Zlt in |- *; intros m n H; elim (Zcompare_Gt_Lt_antisym n m);     auto with arith. Qed. Lemma Zge_le : forall n m:Z, n >= m -> m <= n. Proof.   intros m n; change (~ m < n -> ~ n > m) in |- *; unfold not in |- *;     intros H1 H2; apply H1; apply Zgt_lt; assumption. Qed. Lemma Zle_ge : forall n m:Z, n <= m -> m >= n. Proof.   intros m n; change (~ m > n -> ~ n < m) in |- *; unfold not in |- *;     intros H1 H2; apply H1; apply Zlt_gt; assumption. Qed. Lemma Zle_not_gt : forall n m:Z, n <= m -> ~ n > m. Proof.   trivial. Qed. Lemma Zgt_not_le : forall n m:Z, n > m -> ~ n <= m. Proof.   intros n m H1 H2; apply H2; assumption. Qed. Lemma Zle_not_lt : forall n m:Z, n <= m -> ~ m < n. Proof.   intros n m H1 H2.   assert (H3 := Zlt_gt _ _ H2).   apply Zle_not_gt with n m; assumption. Qed. Lemma Zlt_not_le : forall n m:Z, n < m -> ~ m <= n. Proof.   intros n m H1 H2.   apply Zle_not_lt with m n; assumption. Qed. Lemma Znot_ge_lt : forall n m:Z, ~ n >= m -> n < m. Proof.   unfold Zge, Zlt in |- *; intros x y H; apply dec_not_not;     [ exact (dec_Zlt x y) | assumption ]. Qed. Lemma Znot_lt_ge : forall n m:Z, ~ n < m -> n >= m. Proof.   unfold Zlt, Zge in |- *; auto with arith. Qed. Lemma Znot_gt_le : forall n m:Z, ~ n > m -> n <= m. Proof.   trivial. Qed. Lemma Znot_le_gt : forall n m:Z, ~ n <= m -> n > m. Proof.   unfold Zle, Zgt in |- *; intros x y H; apply dec_not_not;     [ exact (dec_Zgt x y) | assumption ]. Qed. Lemma Zge_iff_le : forall n m:Z, n >= m <-> m <= n. Proof.   intros x y; intros. split. intro. apply Zge_le. assumption.   intro. apply Zle_ge. assumption. Qed. Lemma Zgt_iff_lt : forall n m:Z, n > m <-> m < n. Proof.   intros x y. split. intro. apply Zgt_lt. assumption.   intro. apply Zlt_gt. assumption. Qed. ```

Equivalence and order properties

``` ```
Reflexivity
``` Lemma Zle_refl : forall n:Z, n <= n. Proof.   intros n; unfold Zle in |- *; rewrite (Zcompare_refl n); discriminate. Qed. Lemma Zeq_le : forall n m:Z, n = m -> n <= m. Proof.   intros; rewrite H; apply Zle_refl. Qed. Hint Resolve Zle_refl: zarith. ```
Antisymmetry
``` Lemma Zle_antisym : forall n m:Z, n <= m -> m <= n -> n = m. Proof.   intros n m H1 H2; destruct (Ztrichotomy n m) as [Hlt| [Heq| Hgt]].   absurd (m > n); [ apply Zle_not_gt | apply Zlt_gt ]; assumption.   assumption.   absurd (n > m); [ apply Zle_not_gt | idtac ]; assumption. Qed. ```
Asymmetry
``` Lemma Zgt_asym : forall n m:Z, n > m -> ~ m > n. Proof.   unfold Zgt in |- *; intros n m H; elim (Zcompare_Gt_Lt_antisym n m);     intros H1 H2; rewrite H1; [ discriminate | assumption ]. Qed. Lemma Zlt_asym : forall n m:Z, n < m -> ~ m < n. Proof.   intros n m H H1; assert (H2 : m > n). apply Zlt_gt; assumption.   assert (H3 : n > m). apply Zlt_gt; assumption.   apply Zgt_asym with m n; assumption. Qed. ```
Irreflexivity
``` Lemma Zgt_irrefl : forall n:Z, ~ n > n. Proof.   intros n H; apply (Zgt_asym n n H H). Qed. Lemma Zlt_irrefl : forall n:Z, ~ n < n. Proof.   intros n H; apply (Zlt_asym n n H H). Qed. Lemma Zlt_not_eq : forall n m:Z, n < m -> n <> m. Proof.   unfold not in |- *; intros x y H H0.   rewrite H0 in H.   apply (Zlt_irrefl _ H). Qed. ```
Large = strict or equal
``` Lemma Zlt_le_weak : forall n m:Z, n < m -> n <= m. Proof.   intros n m Hlt; apply Znot_gt_le; apply Zgt_asym; apply Zlt_gt; assumption. Qed. Lemma Zle_lt_or_eq : forall n m:Z, n <= m -> n < m \/ n = m. Proof.   intros n m H; destruct (Ztrichotomy n m) as [Hlt| [Heq| Hgt]];     [ left; assumption       | right; assumption       | absurd (n > m); [ apply Zle_not_gt | idtac ]; assumption ]. Qed. ```
Dichotomy
``` Lemma Zle_or_lt : forall n m:Z, n <= m \/ m < n. Proof.   intros n m; destruct (Ztrichotomy n m) as [Hlt| [Heq| Hgt]];     [ left; apply Znot_gt_le; intro Hgt; assert (Hgt' := Zlt_gt _ _ Hlt);       apply Zgt_asym with m n; assumption       | left; rewrite Heq; apply Zle_refl       | right; apply Zgt_lt; assumption ]. Qed. ```
Transitivity of strict orders
``` Lemma Zgt_trans : forall n m p:Z, n > m -> m > p -> n > p. Proof.   exact Zcompare_Gt_trans. Qed. Lemma Zlt_trans : forall n m p:Z, n < m -> m < p -> n < p. Proof.   intros n m p H1 H2; apply Zgt_lt; apply Zgt_trans with (m := m); apply Zlt_gt;     assumption. Qed. ```
Mixed transitivity
``` Lemma Zle_gt_trans : forall n m p:Z, m <= n -> m > p -> n > p. Proof.   intros n m p H1 H2; destruct (Zle_lt_or_eq m n H1) as [Hlt| Heq];     [ apply Zgt_trans with m; [ apply Zlt_gt; assumption | assumption ]       | rewrite <- Heq; assumption ]. Qed. Lemma Zgt_le_trans : forall n m p:Z, n > m -> p <= m -> n > p. Proof.   intros n m p H1 H2; destruct (Zle_lt_or_eq p m H2) as [Hlt| Heq];     [ apply Zgt_trans with m; [ assumption | apply Zlt_gt; assumption ]       | rewrite Heq; assumption ]. Qed. Lemma Zlt_le_trans : forall n m p:Z, n < m -> m <= p -> n < p.   intros n m p H1 H2; apply Zgt_lt; apply Zle_gt_trans with (m := m);     [ assumption | apply Zlt_gt; assumption ]. Qed. Lemma Zle_lt_trans : forall n m p:Z, n <= m -> m < p -> n < p. Proof.   intros n m p H1 H2; apply Zgt_lt; apply Zgt_le_trans with (m := m);     [ apply Zlt_gt; assumption | assumption ]. Qed. ```
Transitivity of large orders
``` Lemma Zle_trans : forall n m p:Z, n <= m -> m <= p -> n <= p. Proof.   intros n m p H1 H2; apply Znot_gt_le.   intro Hgt; apply Zle_not_gt with n m. assumption.   exact (Zgt_le_trans n p m Hgt H2). Qed. Lemma Zge_trans : forall n m p:Z, n >= m -> m >= p -> n >= p. Proof.   intros n m p H1 H2.   apply Zle_ge.   apply Zle_trans with m; apply Zge_le; trivial. Qed. Hint Resolve Zle_trans: zarith. ```

Compatibility of order and operations on Z

``` ```

Successor

``` ```
Compatibility of successor wrt to order
``` Lemma Zsucc_le_compat : forall n m:Z, m <= n -> Zsucc m <= Zsucc n. Proof.   unfold Zle, not in |- *; intros m n H1 H2; apply H1;     rewrite <- (Zcompare_plus_compat n m 1); do 2 rewrite (Zplus_comm 1);       exact H2. Qed. Lemma Zsucc_gt_compat : forall n m:Z, m > n -> Zsucc m > Zsucc n. Proof.   unfold Zgt in |- *; intros n m H; rewrite Zcompare_succ_compat;     auto with arith. Qed. Lemma Zsucc_lt_compat : forall n m:Z, n < m -> Zsucc n < Zsucc m. Proof.   intros n m H; apply Zgt_lt; apply Zsucc_gt_compat; apply Zlt_gt; assumption. Qed. Hint Resolve Zsucc_le_compat: zarith. ```
Simplification of successor wrt to order
``` Lemma Zsucc_gt_reg : forall n m:Z, Zsucc m > Zsucc n -> m > n. Proof.   unfold Zsucc, Zgt in |- *; intros n p;     do 2 rewrite (fun m:Z => Zplus_comm m 1);       rewrite (Zcompare_plus_compat p n 1); trivial with arith. Qed. Lemma Zsucc_le_reg : forall n m:Z, Zsucc m <= Zsucc n -> m <= n. Proof.   unfold Zle, not in |- *; intros m n H1 H2; apply H1; unfold Zsucc in |- *;     do 2 rewrite <- (Zplus_comm 1); rewrite (Zcompare_plus_compat n m 1);       assumption. Qed. Lemma Zsucc_lt_reg : forall n m:Z, Zsucc n < Zsucc m -> n < m. Proof.   intros n m H; apply Zgt_lt; apply Zsucc_gt_reg; apply Zlt_gt; assumption. Qed. ```
Special base instances of order
``` Lemma Zgt_succ : forall n:Z, Zsucc n > n. Proof.   exact Zcompare_succ_Gt. Qed. Lemma Znot_le_succ : forall n:Z, ~ Zsucc n <= n. Proof.   intros n; apply Zgt_not_le; apply Zgt_succ. Qed. Lemma Zlt_succ : forall n:Z, n < Zsucc n. Proof.   intro n; apply Zgt_lt; apply Zgt_succ. Qed. Lemma Zlt_pred : forall n:Z, Zpred n < n. Proof.   intros n; apply Zsucc_lt_reg; rewrite <- Zsucc_pred; apply Zlt_succ. Qed. ```
Relating strict and large order using successor or predecessor
``` Lemma Zgt_le_succ : forall n m:Z, m > n -> Zsucc n <= m. Proof.   unfold Zgt, Zle in |- *; intros n p H; elim (Zcompare_Gt_not_Lt p n);     intros H1 H2; unfold not in |- *; intros H3; unfold not in H1;       apply H1;         [ assumption | elim (Zcompare_Gt_Lt_antisym (n + 1) p); intros H4 H5; apply H4; exact H3 ]. Qed. Lemma Zlt_gt_succ : forall n m:Z, n <= m -> Zsucc m > n. Proof.   intros n p H; apply Zgt_le_trans with p.   apply Zgt_succ.   assumption. Qed. Lemma Zle_lt_succ : forall n m:Z, n <= m -> n < Zsucc m. Proof.   intros n m H; apply Zgt_lt; apply Zlt_gt_succ; assumption. Qed. Lemma Zlt_le_succ : forall n m:Z, n < m -> Zsucc n <= m. Proof.   intros n p H; apply Zgt_le_succ; apply Zlt_gt; assumption. Qed. Lemma Zgt_succ_le : forall n m:Z, Zsucc m > n -> n <= m. Proof.   intros n p H; apply Zsucc_le_reg; apply Zgt_le_succ; assumption. Qed. Lemma Zlt_succ_le : forall n m:Z, n < Zsucc m -> n <= m. Proof.   intros n m H; apply Zgt_succ_le; apply Zlt_gt; assumption. Qed. Lemma Zlt_succ_gt : forall n m:Z, Zsucc n <= m -> m > n. Proof.   intros n m H; apply Zle_gt_trans with (m := Zsucc n);     [ assumption | apply Zgt_succ ]. Qed. ```
Weakening order
``` Lemma Zle_succ : forall n:Z, n <= Zsucc n. Proof.   intros n; apply Zgt_succ_le; apply Zgt_trans with (m := Zsucc n);     apply Zgt_succ. Qed. Hint Resolve Zle_succ: zarith. Lemma Zle_pred : forall n:Z, Zpred n <= n. Proof.   intros n; pattern n at 2 in |- *; rewrite Zsucc_pred; apply Zle_succ. Qed. Lemma Zlt_lt_succ : forall n m:Z, n < m -> n < Zsucc m.   intros n m H; apply Zgt_lt; apply Zgt_trans with (m := m);     [ apply Zgt_succ | apply Zlt_gt; assumption ]. Qed. Lemma Zle_le_succ : forall n m:Z, n <= m -> n <= Zsucc m. Proof.   intros x y H.   apply Zle_trans with y; trivial with zarith. Qed. Lemma Zle_succ_le : forall n m:Z, Zsucc n <= m -> n <= m. Proof.   intros n m H; apply Zle_trans with (m := Zsucc n);     [ apply Zle_succ | assumption ]. Qed. Hint Resolve Zle_le_succ: zarith. ```
Relating order wrt successor and order wrt predecessor
``` Lemma Zgt_succ_pred : forall n m:Z, m > Zsucc n -> Zpred m > n. Proof.   unfold Zgt, Zsucc, Zpred in |- *; intros n p H;     rewrite <- (fun x y => Zcompare_plus_compat x y 1);       rewrite (Zplus_comm p); rewrite Zplus_assoc;         rewrite (fun x => Zplus_comm x n); simpl in |- *;           assumption. Qed. Lemma Zlt_succ_pred : forall n m:Z, Zsucc n < m -> n < Zpred m. Proof.   intros n p H; apply Zsucc_lt_reg; rewrite <- Zsucc_pred; assumption. Qed. ```
Relating strict order and large order on positive
``` Lemma Zlt_0_le_0_pred : forall n:Z, 0 < n -> 0 <= Zpred n. Proof.   intros x H.   rewrite (Zsucc_pred x) in H.   apply Zgt_succ_le.   apply Zlt_gt.   assumption. Qed. Lemma Zgt_0_le_0_pred : forall n:Z, n > 0 -> 0 <= Zpred n. Proof.   intros; apply Zlt_0_le_0_pred; apply Zgt_lt. assumption. Qed. ```
Special cases of ordered integers
``` Lemma Zlt_0_1 : 0 < 1. Proof.   change (0 < Zsucc 0) in |- *. apply Zlt_succ. Qed. Lemma Zle_0_1 : 0 <= 1. Proof.   change (0 <= Zsucc 0) in |- *. apply Zle_succ. Qed. Lemma Zle_neg_pos : forall p q:positive, Zneg p <= Zpos q. Proof.   intros p; red in |- *; simpl in |- *; red in |- *; intros H; discriminate. Qed. Lemma Zgt_pos_0 : forall p:positive, Zpos p > 0. Proof.   unfold Zgt in |- *; trivial. Qed. Lemma Zle_0_pos : forall p:positive, 0 <= Zpos p. Proof.   intro; unfold Zle in |- *; discriminate. Qed. Lemma Zlt_neg_0 : forall p:positive, Zneg p < 0. Proof.   unfold Zlt in |- *; trivial. Qed. Lemma Zle_0_nat : forall n:nat, 0 <= Z_of_nat n. Proof.   simple induction n; simpl in |- *; intros;     [ apply Zle_refl | unfold Zle in |- *; simpl in |- *; discriminate ]. Qed. Hint Immediate Zeq_le: zarith. ```
Transitivity using successor
``` Lemma Zge_trans_succ : forall n m p:Z, Zsucc n > m -> m > p -> n > p. Proof.   intros n m p H1 H2; apply Zle_gt_trans with (m := m);     [ apply Zgt_succ_le; assumption | assumption ]. Qed. ```
Derived lemma
``` Lemma Zgt_succ_gt_or_eq : forall n m:Z, Zsucc n > m -> n > m \/ m = n. Proof.   intros n m H.   assert (Hle : m <= n).   apply Zgt_succ_le; assumption.   destruct (Zle_lt_or_eq _ _ Hle) as [Hlt| Heq].   left; apply Zlt_gt; assumption.   right; assumption. Qed. ```

``` ```
Compatibility of addition wrt to order
``` Lemma Zplus_gt_compat_l : forall n m p:Z, n > m -> p + n > p + m. Proof.   unfold Zgt in |- *; intros n m p H; rewrite (Zcompare_plus_compat n m p);     assumption. Qed. Lemma Zplus_gt_compat_r : forall n m p:Z, n > m -> n + p > m + p. Proof.   intros n m p H; rewrite (Zplus_comm n p); rewrite (Zplus_comm m p);     apply Zplus_gt_compat_l; trivial. Qed. Lemma Zplus_le_compat_l : forall n m p:Z, n <= m -> p + n <= p + m. Proof.   intros n m p; unfold Zle, not in |- *; intros H1 H2; apply H1;     rewrite <- (Zcompare_plus_compat n m p); assumption. Qed. Lemma Zplus_le_compat_r : forall n m p:Z, n <= m -> n + p <= m + p. Proof.   intros a b c; do 2 rewrite (fun n:Z => Zplus_comm n c);     exact (Zplus_le_compat_l a b c). Qed. Lemma Zplus_lt_compat_l : forall n m p:Z, n < m -> p + n < p + m. Proof.   unfold Zlt in |- *; intros n m p; rewrite Zcompare_plus_compat;     trivial with arith. Qed. Lemma Zplus_lt_compat_r : forall n m p:Z, n < m -> n + p < m + p. Proof.   intros n m p H; rewrite (Zplus_comm n p); rewrite (Zplus_comm m p);     apply Zplus_lt_compat_l; trivial. Qed. Lemma Zplus_lt_le_compat : forall n m p q:Z, n < m -> p <= q -> n + p < m + q. Proof.   intros a b c d H0 H1.   apply Zlt_le_trans with (b + c).   apply Zplus_lt_compat_r; trivial.   apply Zplus_le_compat_l; trivial. Qed. Lemma Zplus_le_lt_compat : forall n m p q:Z, n <= m -> p < q -> n + p < m + q. Proof.   intros a b c d H0 H1.   apply Zle_lt_trans with (b + c).   apply Zplus_le_compat_r; trivial.   apply Zplus_lt_compat_l; trivial. Qed. Lemma Zplus_le_compat : forall n m p q:Z, n <= m -> p <= q -> n + p <= m + q. Proof.   intros n m p q; intros H1 H2; apply Zle_trans with (m := n + q);     [ apply Zplus_le_compat_l; assumption | apply Zplus_le_compat_r; assumption ]. Qed. Lemma Zplus_lt_compat : forall n m p q:Z, n < m -> p < q -> n + p < m + q.   intros; apply Zplus_le_lt_compat. apply Zlt_le_weak; assumption. assumption. Qed. ```
Compatibility of addition wrt to being positive
``` Lemma Zplus_le_0_compat : forall n m:Z, 0 <= n -> 0 <= m -> 0 <= n + m. Proof.   intros x y H1 H2; rewrite <- (Zplus_0_l 0); apply Zplus_le_compat; assumption. Qed. ```
Simplification of addition wrt to order
``` Lemma Zplus_gt_reg_l : forall n m p:Z, p + n > p + m -> n > m. Proof.   unfold Zgt in |- *; intros n m p H; rewrite <- (Zcompare_plus_compat n m p);     assumption. Qed. Lemma Zplus_gt_reg_r : forall n m p:Z, n + p > m + p -> n > m. Proof.   intros n m p H; apply Zplus_gt_reg_l with p.   rewrite (Zplus_comm p n); rewrite (Zplus_comm p m); trivial. Qed. Lemma Zplus_le_reg_l : forall n m p:Z, p + n <= p + m -> n <= m. Proof.   intros n m p; unfold Zle, not in |- *; intros H1 H2; apply H1;     rewrite (Zcompare_plus_compat n m p); assumption. Qed. Lemma Zplus_le_reg_r : forall n m p:Z, n + p <= m + p -> n <= m. Proof.   intros n m p H; apply Zplus_le_reg_l with p.   rewrite (Zplus_comm p n); rewrite (Zplus_comm p m); trivial. Qed. Lemma Zplus_lt_reg_l : forall n m p:Z, p + n < p + m -> n < m. Proof.   unfold Zlt in |- *; intros n m p; rewrite Zcompare_plus_compat;     trivial with arith. Qed. Lemma Zplus_lt_reg_r : forall n m p:Z, n + p < m + p -> n < m. Proof.   intros n m p H; apply Zplus_lt_reg_l with p.   rewrite (Zplus_comm p n); rewrite (Zplus_comm p m); trivial. Qed. ```

Multiplication

``` ```
Compatibility of multiplication by a positive wrt to order
``` Lemma Zmult_le_compat_r : forall n m p:Z, n <= m -> 0 <= p -> n * p <= m * p. Proof.   intros a b c H H0; destruct c.   do 2 rewrite Zmult_0_r; assumption.   rewrite (Zmult_comm a); rewrite (Zmult_comm b).   unfold Zle in |- *; rewrite Zcompare_mult_compat; assumption.   unfold Zle in H0; contradiction H0; reflexivity. Qed. Lemma Zmult_le_compat_l : forall n m p:Z, n <= m -> 0 <= p -> p * n <= p * m. Proof.   intros a b c H1 H2; rewrite (Zmult_comm c a); rewrite (Zmult_comm c b).   apply Zmult_le_compat_r; trivial. Qed. Lemma Zmult_lt_compat_r : forall n m p:Z, 0 < p -> n < m -> n * p < m * p. Proof.   intros x y z H H0; destruct z.   contradiction (Zlt_irrefl 0).   rewrite (Zmult_comm x); rewrite (Zmult_comm y).   unfold Zlt in |- *; rewrite Zcompare_mult_compat; assumption.   discriminate H. Qed. Lemma Zmult_gt_compat_r : forall n m p:Z, p > 0 -> n > m -> n * p > m * p. Proof.   intros x y z; intros; apply Zlt_gt; apply Zmult_lt_compat_r; apply Zgt_lt;     assumption. Qed. Lemma Zmult_gt_0_lt_compat_r :   forall n m p:Z, p > 0 -> n < m -> n * p < m * p. Proof.   intros x y z; intros; apply Zmult_lt_compat_r;     [ apply Zgt_lt; assumption | assumption ]. Qed. Lemma Zmult_gt_0_le_compat_r :   forall n m p:Z, p > 0 -> n <= m -> n * p <= m * p. Proof.   intros x y z Hz Hxy.   elim (Zle_lt_or_eq x y Hxy).   intros; apply Zlt_le_weak.   apply Zmult_gt_0_lt_compat_r; trivial.   intros; apply Zeq_le.   rewrite H; trivial. Qed. Lemma Zmult_lt_0_le_compat_r :   forall n m p:Z, 0 < p -> n <= m -> n * p <= m * p. Proof.   intros x y z; intros; apply Zmult_gt_0_le_compat_r; try apply Zlt_gt;     assumption. Qed. Lemma Zmult_gt_0_lt_compat_l :   forall n m p:Z, p > 0 -> n < m -> p * n < p * m. Proof.   intros x y z; intros.   rewrite (Zmult_comm z x); rewrite (Zmult_comm z y);     apply Zmult_gt_0_lt_compat_r; assumption. Qed. Lemma Zmult_lt_compat_l : forall n m p:Z, 0 < p -> n < m -> p * n < p * m. Proof.   intros x y z; intros.   rewrite (Zmult_comm z x); rewrite (Zmult_comm z y);     apply Zmult_gt_0_lt_compat_r; try apply Zlt_gt; assumption. Qed. Lemma Zmult_gt_compat_l : forall n m p:Z, p > 0 -> n > m -> p * n > p * m. Proof.   intros x y z; intros; rewrite (Zmult_comm z x); rewrite (Zmult_comm z y);     apply Zmult_gt_compat_r; assumption. Qed. Lemma Zmult_ge_compat_r : forall n m p:Z, n >= m -> p >= 0 -> n * p >= m * p. Proof.   intros a b c H1 H2; apply Zle_ge.   apply Zmult_le_compat_r; apply Zge_le; trivial. Qed. Lemma Zmult_ge_compat_l : forall n m p:Z, n >= m -> p >= 0 -> p * n >= p * m. Proof.   intros a b c H1 H2; apply Zle_ge.   apply Zmult_le_compat_l; apply Zge_le; trivial. Qed. Lemma Zmult_ge_compat :   forall n m p q:Z, n >= p -> m >= q -> p >= 0 -> q >= 0 -> n * m >= p * q. Proof.   intros a b c d H0 H1 H2 H3.   apply Zge_trans with (a * d).   apply Zmult_ge_compat_l; trivial.   apply Zge_trans with c; trivial.   apply Zmult_ge_compat_r; trivial. Qed. Lemma Zmult_le_compat :   forall n m p q:Z, n <= p -> m <= q -> 0 <= n -> 0 <= m -> n * m <= p * q. Proof.   intros a b c d H0 H1 H2 H3.   apply Zle_trans with (c * b).   apply Zmult_le_compat_r; assumption.   apply Zmult_le_compat_l.   assumption.   apply Zle_trans with a; assumption. Qed. ```
Simplification of multiplication by a positive wrt to being positive
``` Lemma Zmult_gt_0_lt_reg_r : forall n m p:Z, p > 0 -> n * p < m * p -> n < m. Proof.   intros x y z; intros; destruct z.   contradiction (Zgt_irrefl 0).   rewrite (Zmult_comm x) in H0; rewrite (Zmult_comm y) in H0.   unfold Zlt in H0; rewrite Zcompare_mult_compat in H0; assumption.   discriminate H. Qed. Lemma Zmult_lt_reg_r : forall n m p:Z, 0 < p -> n * p < m * p -> n < m. Proof.   intros a b c H0 H1.   apply Zmult_gt_0_lt_reg_r with c; try apply Zlt_gt; assumption. Qed. Lemma Zmult_le_reg_r : forall n m p:Z, p > 0 -> n * p <= m * p -> n <= m. Proof.   intros x y z Hz Hxy.   elim (Zle_lt_or_eq (x * z) (y * z) Hxy).   intros; apply Zlt_le_weak.   apply Zmult_gt_0_lt_reg_r with z; trivial.   intros; apply Zeq_le.   apply Zmult_reg_r with z.   intro. rewrite H0 in Hz. contradiction (Zgt_irrefl 0).   assumption. Qed. Lemma Zmult_lt_0_le_reg_r : forall n m p:Z, 0 < p -> n * p <= m * p -> n <= m. Proof.   intros x y z; intros; apply Zmult_le_reg_r with z.   try apply Zlt_gt; assumption.   assumption. Qed. Lemma Zmult_ge_reg_r : forall n m p:Z, p > 0 -> n * p >= m * p -> n >= m. Proof.   intros a b c H1 H2; apply Zle_ge; apply Zmult_le_reg_r with c; trivial.   apply Zge_le; trivial. Qed. Lemma Zmult_gt_reg_r : forall n m p:Z, p > 0 -> n * p > m * p -> n > m. Proof.   intros a b c H1 H2; apply Zlt_gt; apply Zmult_gt_0_lt_reg_r with c; trivial.   apply Zgt_lt; trivial. Qed. ```
Compatibility of multiplication by a positive wrt to being positive
``` Lemma Zmult_le_0_compat : forall n m:Z, 0 <= n -> 0 <= m -> 0 <= n * m. Proof.   intros x y; case x.   intros; rewrite Zmult_0_l; trivial.   intros p H1; unfold Zle in |- *.   pattern 0 at 2 in |- *; rewrite <- (Zmult_0_r (Zpos p)).   rewrite Zcompare_mult_compat; trivial.   intros p H1 H2; absurd (0 > Zneg p); trivial.   unfold Zgt in |- *; simpl in |- *; auto with zarith. Qed. Lemma Zmult_gt_0_compat : forall n m:Z, n > 0 -> m > 0 -> n * m > 0. Proof.   intros x y; case x.   intros H; discriminate H.   intros p H1; unfold Zgt in |- *; pattern 0 at 2 in |- *;     rewrite <- (Zmult_0_r (Zpos p)).   rewrite Zcompare_mult_compat; trivial.   intros p H; discriminate H. Qed. Lemma Zmult_lt_0_compat : forall n m:Z, 0 < n -> 0 < m -> 0 < n * m. Proof.   intros a b apos bpos.   apply Zgt_lt.   apply Zmult_gt_0_compat; try apply Zlt_gt; assumption. Qed. ```
For compatibility
``` Notation Zmult_lt_O_compat := Zmult_lt_0_compat (only parsing). Lemma Zmult_gt_0_le_0_compat : forall n m:Z, n > 0 -> 0 <= m -> 0 <= m * n. Proof.   intros x y H1 H2; apply Zmult_le_0_compat; trivial.   apply Zlt_le_weak; apply Zgt_lt; trivial. Qed. ```
Simplification of multiplication by a positive wrt to being positive
``` Lemma Zmult_le_0_reg_r : forall n m:Z, n > 0 -> 0 <= m * n -> 0 <= m. Proof.   intros x y; case x;     [ simpl in |- *; unfold Zgt in |- *; simpl in |- *; intros H; discriminate H | intros p H1; unfold Zle in |- *; rewrite Zmult_comm; pattern 0 at 1 in |- *; rewrite <- (Zmult_0_r (Zpos p)); rewrite Zcompare_mult_compat; auto with arith | intros p; unfold Zgt in |- *; simpl in |- *; intros H; discriminate H ]. Qed. Lemma Zmult_gt_0_lt_0_reg_r : forall n m:Z, n > 0 -> 0 < m * n -> 0 < m. Proof.   intros x y; case x;     [ simpl in |- *; unfold Zgt in |- *; simpl in |- *; intros H; discriminate H | intros p H1; unfold Zlt in |- *; rewrite Zmult_comm; pattern 0 at 1 in |- *; rewrite <- (Zmult_0_r (Zpos p)); rewrite Zcompare_mult_compat; auto with arith | intros p; unfold Zgt in |- *; simpl in |- *; intros H; discriminate H ]. Qed. Lemma Zmult_lt_0_reg_r : forall n m:Z, 0 < n -> 0 < m * n -> 0 < m. Proof.   intros x y; intros; eapply Zmult_gt_0_lt_0_reg_r with x; try apply Zlt_gt;     assumption. Qed. Lemma Zmult_gt_0_reg_l : forall n m:Z, n > 0 -> n * m > 0 -> m > 0. Proof.   intros x y; case x.   intros H; discriminate H.   intros p H1; unfold Zgt in |- *.   pattern 0 at 1 in |- *; rewrite <- (Zmult_0_r (Zpos p)).   rewrite Zcompare_mult_compat; trivial.   intros p H; discriminate H. Qed. ```

Square

``` ```
Simplification of square wrt order
``` Lemma Zgt_square_simpl :   forall n m:Z, n >= 0 -> n * n > m * m -> n > m. Proof.   intros n m H0 H1.   case (dec_Zlt m n).   intro; apply Zlt_gt; trivial.   intros H2; cut (m >= n).   intros H.   elim Zgt_not_le with (1 := H1).   apply Zge_le.   apply Zmult_ge_compat; auto.   apply Znot_lt_ge; trivial. Qed. Lemma Zlt_square_simpl :   forall n m:Z, 0 <= n -> m * m < n * n -> m < n. Proof.   intros x y H0 H1.   apply Zgt_lt.   apply Zgt_square_simpl; try apply Zle_ge; try apply Zlt_gt; assumption. Qed. ```

Equivalence between inequalities

``` Lemma Zle_plus_swap : forall n m p:Z, n + p <= m <-> n <= m - p. Proof.   intros x y z; intros. split. intro. rewrite <- (Zplus_0_r x). rewrite <- (Zplus_opp_r z).   rewrite Zplus_assoc. exact (Zplus_le_compat_r _ _ _ H).   intro. rewrite <- (Zplus_0_r y). rewrite <- (Zplus_opp_l z). rewrite Zplus_assoc.   apply Zplus_le_compat_r. assumption. Qed. Lemma Zlt_plus_swap : forall n m p:Z, n + p < m <-> n < m - p. Proof.   intros x y z; intros. split. intro. unfold Zminus in |- *. rewrite Zplus_comm. rewrite <- (Zplus_0_l x).   rewrite <- (Zplus_opp_l z). rewrite Zplus_assoc_reverse. apply Zplus_lt_compat_l. rewrite Zplus_comm.   assumption.   intro. rewrite Zplus_comm. rewrite <- (Zplus_0_l y). rewrite <- (Zplus_opp_r z).   rewrite Zplus_assoc_reverse. apply Zplus_lt_compat_l. rewrite Zplus_comm. assumption. Qed. Lemma Zeq_plus_swap : forall n m p:Z, n + p = m <-> n = m - p. Proof.   intros x y z; intros. split. intro. apply Zplus_minus_eq. symmetry in |- *. rewrite Zplus_comm.   assumption.   intro. rewrite H. unfold Zminus in |- *. rewrite Zplus_assoc_reverse.   rewrite Zplus_opp_l. apply Zplus_0_r. Qed. Lemma Zlt_minus_simpl_swap : forall n m:Z, 0 < m -> n - m < n. Proof.   intros n m H; apply Zplus_lt_reg_l with (p := m); rewrite Zplus_minus;     pattern n at 1 in |- *; rewrite <- (Zplus_0_r n);       rewrite (Zplus_comm m n); apply Zplus_lt_compat_l;         assumption. Qed. Lemma Zlt_0_minus_lt : forall n m:Z, 0 < n - m -> m < n. Proof.   intros n m H; apply Zplus_lt_reg_l with (p := - m); rewrite Zplus_opp_l;     rewrite Zplus_comm; exact H. Qed. Lemma Zle_0_minus_le : forall n m:Z, 0 <= n - m -> m <= n. Proof.   intros n m H; apply Zplus_le_reg_l with (p := - m); rewrite Zplus_opp_l;     rewrite Zplus_comm; exact H. Qed. Lemma Zle_minus_le_0 : forall n m:Z, m <= n -> 0 <= n - m. Proof.   intros n m H; unfold Zminus; apply Zplus_le_reg_r with (p := m);     rewrite <- Zplus_assoc; rewrite Zplus_opp_l; rewrite Zplus_0_r; exact H. Qed. ```
For compatibility
``` Notation Zlt_O_minus_lt := Zlt_0_minus_lt (only parsing). ```