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Diffstat (limited to 'theories/ZArith/Zorder.v')
| -rw-r--r-- | theories/ZArith/Zorder.v | 862 |
1 files changed, 440 insertions, 422 deletions
diff --git a/theories/ZArith/Zorder.v b/theories/ZArith/Zorder.v index b81cc580..47490be6 100644 --- a/theories/ZArith/Zorder.v +++ b/theories/ZArith/Zorder.v @@ -5,13 +5,13 @@ (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: Zorder.v 6983 2005-05-02 10:47:51Z herbelin $ i*) +(*i $Id: Zorder.v 9302 2006-10-27 21:21:17Z barras $ i*) (** Binary Integers (Pierre Crégut (CNET, Lannion, France) *) Require Import BinPos. Require Import BinInt. -Require Import Arith. +Require Import Arith_base. Require Import Decidable. Require Import Zcompare. @@ -19,178 +19,180 @@ Open Local Scope Z_scope. Implicit Types x y z : Z. -(**********************************************************************) +(*********************************************************) (** Properties of the order relations on binary integers *) -(** Trichotomy *) +(** * 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)). + 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. + [ 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. + [ left | right; left | right; right ]; assumption. Qed. (**********************************************************************) -(** Decidability of equality and order on Z *) +(** * 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 ]. + 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 ] ]. + 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 ]. + 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 ]. + 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 ]. + 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 ]. + 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 ] ]. + 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 *) +(** * 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. + 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. + 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. + 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. + 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. + trivial. Qed. Lemma Zgt_not_le : forall n m:Z, n > m -> ~ n <= m. Proof. -intros n m H1 H2; apply H2; assumption. + 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. + 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. + 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 ]. + 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. + unfold Zlt, Zge in |- *; auto with arith. Qed. Lemma Znot_gt_le : forall n m:Z, ~ n > m -> n <= m. Proof. -trivial. + 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 ]. + 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. + 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. + 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. + 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. + intros; rewrite H; apply Zle_refl. Qed. Hint Resolve Zle_refl: zarith. @@ -199,7 +201,7 @@ Hint Resolve Zle_refl: zarith. 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]]. + 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. @@ -209,138 +211,143 @@ Qed. 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 ]. + 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. + 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). + 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). + 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). + 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. + 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 ]. + 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 ]. + 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. + 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. + 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 ]. + 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 ]. + 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 ]. + 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 ]. + 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). + 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. + 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. + 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. + 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. + intros n m H; apply Zgt_lt; apply Zsucc_gt_compat; apply Zlt_gt; assumption. Qed. Hint Resolve Zsucc_le_compat: zarith. @@ -349,231 +356,119 @@ Hint Resolve Zsucc_le_compat: zarith. 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. + 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. + 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. - -(** 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 ]. + intros n m H; apply Zgt_lt; apply Zsucc_gt_reg; apply Zlt_gt; 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. - (** Special base instances of order *) Lemma Zgt_succ : forall n:Z, Zsucc n > n. Proof. -exact Zcompare_succ_Gt. + 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. + 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. + 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. + 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 ]. + 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. + 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. + 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. + 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. + 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. + 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 ]. + 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. + 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. + 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 ]. + 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. + 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 ]. + intros n m H; apply Zle_trans with (m := Zsucc n); + [ apply Zle_succ | assumption ]. Qed. Hint Resolve Zle_le_succ: zarith. @@ -582,31 +477,32 @@ Hint Resolve Zle_le_succ: zarith. 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. + 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. + 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. -intros x H. -rewrite (Zsucc_pred x) in H. -apply Zgt_succ_le. -apply Zlt_gt. -assumption. +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. -intros; apply Zlt_0_le_0_pred; apply Zgt_lt. assumption. +Proof. + intros; apply Zlt_0_le_0_pred; apply Zgt_lt. assumption. Qed. @@ -614,35 +510,39 @@ Qed. Lemma Zlt_0_1 : 0 < 1. Proof. -change (0 < Zsucc 0) in |- *. apply Zlt_succ. + change (0 < Zsucc 0) in |- *. apply Zlt_succ. Qed. Lemma Zle_0_1 : 0 <= 1. Proof. -change (0 <= Zsucc 0) in |- *. apply Zle_succ. + 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. + intros p; red in |- *; simpl in |- *; red in |- *; intros H; discriminate. Qed. Lemma Zgt_pos_0 : forall p:positive, Zpos p > 0. -unfold Zgt in |- *; trivial. +Proof. + unfold Zgt in |- *; trivial. Qed. - (* weaker but useful (in [Zpower] for instance) *) +(* weaker but useful (in [Zpower] for instance) *) Lemma Zle_0_pos : forall p:positive, 0 <= Zpos p. -intro; unfold Zle in |- *; discriminate. +Proof. + intro; unfold Zle in |- *; discriminate. Qed. Lemma Zlt_neg_0 : forall p:positive, Zneg p < 0. -unfold Zlt in |- *; trivial. +Proof. + unfold Zlt in |- *; trivial. Qed. Lemma Zle_0_nat : forall n:nat, 0 <= Z_of_nat n. -simple induction n; simpl in |- *; intros; - [ apply Zle_refl | unfold Zle in |- *; simpl in |- *; discriminate ]. +Proof. + simple induction n; simpl in |- *; intros; + [ apply Zle_refl | unfold Zle in |- *; simpl in |- *; discriminate ]. Qed. Hint Immediate Zeq_le: zarith. @@ -651,178 +551,294 @@ Hint Immediate Zeq_le: zarith. 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 ]. + 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). + intros n m H. + assert (Hle : m <= n). apply Zgt_succ_le; assumption. -destruct (Zle_lt_or_eq _ _ Hle) as [Hlt| Heq]. + destruct (Zle_lt_or_eq _ _ Hle) as [Hlt| Heq]. left; apply Zlt_gt; assumption. right; assumption. Qed. -(** Compatibility of multiplication by a positive wrt to order *) +(** ** Addition *) +(** 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. + 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 |- *; 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. + 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. + 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. + 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. + 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. + 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 ]. + 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. + 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. + 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. + 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. + 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. + 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. + 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. + 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. + 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. + 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. + 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. + 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. + 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. + 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. + 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. + 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. + 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. + 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. + 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. + assumption. Qed. Lemma Zmult_lt_0_le_reg_r : forall n m p:Z, 0 < p -> n * p <= m * p -> n <= m. -intros x y z; intros; apply Zmult_le_reg_r with z. -try apply Zlt_gt; assumption. -assumption. +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. -intros a b c H1 H2; apply Zle_ge; apply Zmult_le_reg_r with c; trivial. -apply Zge_le; trivial. +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. -intros a b c H1 H2; apply Zlt_gt; apply Zmult_gt_0_lt_reg_r with c; trivial. -apply Zgt_lt; trivial. +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. @@ -830,154 +846,156 @@ Qed. 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 |- *. + 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. + 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)). + 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. + intros p H; discriminate H. Qed. Lemma Zmult_lt_0_compat : forall n m:Z, 0 < n -> 0 < m -> 0 < n * m. -intros a b apos bpos. -apply Zgt_lt. -apply Zmult_gt_0_compat; try apply Zlt_gt; assumption. +Proof. + intros a b apos bpos. + apply Zgt_lt. + apply Zmult_gt_0_compat; try apply Zlt_gt; assumption. Qed. -(* For compatibility *) +(** 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. + 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 ]. + 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 ]. + 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. + 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. + 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. + 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. + 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. + 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. + 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 *) +(** * 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. + 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. + 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. + 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. + 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. + 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. + 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. + 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. + 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 *) +(** For compatibility *) Notation Zlt_O_minus_lt := Zlt_0_minus_lt (only parsing). |