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Diffstat (limited to 'theories/Numbers/Integer/Abstract/ZMulOrder.v')
| -rw-r--r-- | theories/Numbers/Integer/Abstract/ZMulOrder.v | 356 |
1 files changed, 123 insertions, 233 deletions
diff --git a/theories/Numbers/Integer/Abstract/ZMulOrder.v b/theories/Numbers/Integer/Abstract/ZMulOrder.v index c7996ffd..99be58eb 100644 --- a/theories/Numbers/Integer/Abstract/ZMulOrder.v +++ b/theories/Numbers/Integer/Abstract/ZMulOrder.v @@ -8,335 +8,225 @@ (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) -(*i $Id: ZMulOrder.v 11674 2008-12-12 19:48:40Z letouzey $ i*) +(*i $Id$ i*) Require Export ZAddOrder. -Module ZMulOrderPropFunct (Import ZAxiomsMod : ZAxiomsSig). -Module Export ZAddOrderPropMod := ZAddOrderPropFunct ZAxiomsMod. -Open Local Scope IntScope. +Module Type ZMulOrderPropFunct (Import Z : ZAxiomsSig'). +Include ZAddOrderPropFunct Z. -Theorem Zmul_lt_pred : - forall p q n m : Z, S p == q -> (p * n < p * m <-> q * n + m < q * m + n). -Proof NZmul_lt_pred. +Local Notation "- 1" := (-(1)). -Theorem Zmul_lt_mono_pos_l : forall p n m : Z, 0 < p -> (n < m <-> p * n < p * m). -Proof NZmul_lt_mono_pos_l. - -Theorem Zmul_lt_mono_pos_r : forall p n m : Z, 0 < p -> (n < m <-> n * p < m * p). -Proof NZmul_lt_mono_pos_r. - -Theorem Zmul_lt_mono_neg_l : forall p n m : Z, p < 0 -> (n < m <-> p * m < p * n). -Proof NZmul_lt_mono_neg_l. - -Theorem Zmul_lt_mono_neg_r : forall p n m : Z, p < 0 -> (n < m <-> m * p < n * p). -Proof NZmul_lt_mono_neg_r. - -Theorem Zmul_le_mono_nonneg_l : forall n m p : Z, 0 <= p -> n <= m -> p * n <= p * m. -Proof NZmul_le_mono_nonneg_l. - -Theorem Zmul_le_mono_nonpos_l : forall n m p : Z, p <= 0 -> n <= m -> p * m <= p * n. -Proof NZmul_le_mono_nonpos_l. - -Theorem Zmul_le_mono_nonneg_r : forall n m p : Z, 0 <= p -> n <= m -> n * p <= m * p. -Proof NZmul_le_mono_nonneg_r. - -Theorem Zmul_le_mono_nonpos_r : forall n m p : Z, p <= 0 -> n <= m -> m * p <= n * p. -Proof NZmul_le_mono_nonpos_r. - -Theorem Zmul_cancel_l : forall n m p : Z, p ~= 0 -> (p * n == p * m <-> n == m). -Proof NZmul_cancel_l. - -Theorem Zmul_cancel_r : forall n m p : Z, p ~= 0 -> (n * p == m * p <-> n == m). -Proof NZmul_cancel_r. - -Theorem Zmul_id_l : forall n m : Z, m ~= 0 -> (n * m == m <-> n == 1). -Proof NZmul_id_l. - -Theorem Zmul_id_r : forall n m : Z, n ~= 0 -> (n * m == n <-> m == 1). -Proof NZmul_id_r. - -Theorem Zmul_le_mono_pos_l : forall n m p : Z, 0 < p -> (n <= m <-> p * n <= p * m). -Proof NZmul_le_mono_pos_l. - -Theorem Zmul_le_mono_pos_r : forall n m p : Z, 0 < p -> (n <= m <-> n * p <= m * p). -Proof NZmul_le_mono_pos_r. - -Theorem Zmul_le_mono_neg_l : forall n m p : Z, p < 0 -> (n <= m <-> p * m <= p * n). -Proof NZmul_le_mono_neg_l. - -Theorem Zmul_le_mono_neg_r : forall n m p : Z, p < 0 -> (n <= m <-> m * p <= n * p). -Proof NZmul_le_mono_neg_r. - -Theorem Zmul_lt_mono_nonneg : - forall n m p q : Z, 0 <= n -> n < m -> 0 <= p -> p < q -> n * p < m * q. -Proof NZmul_lt_mono_nonneg. - -Theorem Zmul_lt_mono_nonpos : - forall n m p q : Z, m <= 0 -> n < m -> q <= 0 -> p < q -> m * q < n * p. +Theorem mul_lt_mono_nonpos : + forall n m p q, m <= 0 -> n < m -> q <= 0 -> p < q -> m * q < n * p. Proof. intros n m p q H1 H2 H3 H4. -apply Zle_lt_trans with (m * p). -apply Zmul_le_mono_nonpos_l; [assumption | now apply Zlt_le_incl]. -apply -> Zmul_lt_mono_neg_r; [assumption | now apply Zlt_le_trans with q]. +apply le_lt_trans with (m * p). +apply mul_le_mono_nonpos_l; [assumption | now apply lt_le_incl]. +apply -> mul_lt_mono_neg_r; [assumption | now apply lt_le_trans with q]. Qed. -Theorem Zmul_le_mono_nonneg : - forall n m p q : Z, 0 <= n -> n <= m -> 0 <= p -> p <= q -> n * p <= m * q. -Proof NZmul_le_mono_nonneg. - -Theorem Zmul_le_mono_nonpos : - forall n m p q : Z, m <= 0 -> n <= m -> q <= 0 -> p <= q -> m * q <= n * p. +Theorem mul_le_mono_nonpos : + forall n m p q, m <= 0 -> n <= m -> q <= 0 -> p <= q -> m * q <= n * p. Proof. intros n m p q H1 H2 H3 H4. -apply Zle_trans with (m * p). -now apply Zmul_le_mono_nonpos_l. -apply Zmul_le_mono_nonpos_r; [now apply Zle_trans with q | assumption]. -Qed. - -Theorem Zmul_pos_pos : forall n m : Z, 0 < n -> 0 < m -> 0 < n * m. -Proof NZmul_pos_pos. - -Theorem Zmul_neg_neg : forall n m : Z, n < 0 -> m < 0 -> 0 < n * m. -Proof NZmul_neg_neg. - -Theorem Zmul_pos_neg : forall n m : Z, 0 < n -> m < 0 -> n * m < 0. -Proof NZmul_pos_neg. - -Theorem Zmul_neg_pos : forall n m : Z, n < 0 -> 0 < m -> n * m < 0. -Proof NZmul_neg_pos. - -Theorem Zmul_nonneg_nonneg : forall n m : Z, 0 <= n -> 0 <= m -> 0 <= n * m. -Proof. -intros n m H1 H2. -rewrite <- (Zmul_0_l m). now apply Zmul_le_mono_nonneg_r. +apply le_trans with (m * p). +now apply mul_le_mono_nonpos_l. +apply mul_le_mono_nonpos_r; [now apply le_trans with q | assumption]. Qed. -Theorem Zmul_nonpos_nonpos : forall n m : Z, n <= 0 -> m <= 0 -> 0 <= n * m. +Theorem mul_nonpos_nonpos : forall n m, n <= 0 -> m <= 0 -> 0 <= n * m. Proof. intros n m H1 H2. -rewrite <- (Zmul_0_l m). now apply Zmul_le_mono_nonpos_r. +rewrite <- (mul_0_l m). now apply mul_le_mono_nonpos_r. Qed. -Theorem Zmul_nonneg_nonpos : forall n m : Z, 0 <= n -> m <= 0 -> n * m <= 0. +Theorem mul_nonneg_nonpos : forall n m, 0 <= n -> m <= 0 -> n * m <= 0. Proof. intros n m H1 H2. -rewrite <- (Zmul_0_l m). now apply Zmul_le_mono_nonpos_r. +rewrite <- (mul_0_l m). now apply mul_le_mono_nonpos_r. Qed. -Theorem Zmul_nonpos_nonneg : forall n m : Z, n <= 0 -> 0 <= m -> n * m <= 0. +Theorem mul_nonpos_nonneg : forall n m, n <= 0 -> 0 <= m -> n * m <= 0. Proof. -intros; rewrite Zmul_comm; now apply Zmul_nonneg_nonpos. +intros; rewrite mul_comm; now apply mul_nonneg_nonpos. Qed. -Theorem Zlt_1_mul_pos : forall n m : Z, 1 < n -> 0 < m -> 1 < n * m. -Proof NZlt_1_mul_pos. - -Theorem Zeq_mul_0 : forall n m : Z, n * m == 0 <-> n == 0 \/ m == 0. -Proof NZeq_mul_0. - -Theorem Zneq_mul_0 : forall n m : Z, n ~= 0 /\ m ~= 0 <-> n * m ~= 0. -Proof NZneq_mul_0. - -Theorem Zeq_square_0 : forall n : Z, n * n == 0 <-> n == 0. -Proof NZeq_square_0. +Notation mul_pos := lt_0_mul (only parsing). -Theorem Zeq_mul_0_l : forall n m : Z, n * m == 0 -> m ~= 0 -> n == 0. -Proof NZeq_mul_0_l. - -Theorem Zeq_mul_0_r : forall n m : Z, n * m == 0 -> n ~= 0 -> m == 0. -Proof NZeq_mul_0_r. - -Theorem Zlt_0_mul : forall n m : Z, 0 < n * m <-> 0 < n /\ 0 < m \/ m < 0 /\ n < 0. -Proof NZlt_0_mul. - -Notation Zmul_pos := Zlt_0_mul (only parsing). - -Theorem Zlt_mul_0 : - forall n m : Z, n * m < 0 <-> n < 0 /\ m > 0 \/ n > 0 /\ m < 0. +Theorem lt_mul_0 : + forall n m, n * m < 0 <-> n < 0 /\ m > 0 \/ n > 0 /\ m < 0. Proof. intros n m; split; [intro H | intros [[H1 H2] | [H1 H2]]]. -destruct (Zlt_trichotomy n 0) as [H1 | [H1 | H1]]; -[| rewrite H1 in H; rewrite Zmul_0_l in H; false_hyp H Zlt_irrefl |]; -(destruct (Zlt_trichotomy m 0) as [H2 | [H2 | H2]]; -[| rewrite H2 in H; rewrite Zmul_0_r in H; false_hyp H Zlt_irrefl |]); +destruct (lt_trichotomy n 0) as [H1 | [H1 | H1]]; +[| rewrite H1 in H; rewrite mul_0_l in H; false_hyp H lt_irrefl |]; +(destruct (lt_trichotomy m 0) as [H2 | [H2 | H2]]; +[| rewrite H2 in H; rewrite mul_0_r in H; false_hyp H lt_irrefl |]); try (left; now split); try (right; now split). -assert (H3 : n * m > 0) by now apply Zmul_neg_neg. -elimtype False; now apply (Zlt_asymm (n * m) 0). -assert (H3 : n * m > 0) by now apply Zmul_pos_pos. -elimtype False; now apply (Zlt_asymm (n * m) 0). -now apply Zmul_neg_pos. now apply Zmul_pos_neg. +assert (H3 : n * m > 0) by now apply mul_neg_neg. +exfalso; now apply (lt_asymm (n * m) 0). +assert (H3 : n * m > 0) by now apply mul_pos_pos. +exfalso; now apply (lt_asymm (n * m) 0). +now apply mul_neg_pos. now apply mul_pos_neg. Qed. -Notation Zmul_neg := Zlt_mul_0 (only parsing). +Notation mul_neg := lt_mul_0 (only parsing). -Theorem Zle_0_mul : - forall n m : Z, 0 <= n * m -> 0 <= n /\ 0 <= m \/ n <= 0 /\ m <= 0. +Theorem le_0_mul : + forall n m, 0 <= n * m -> 0 <= n /\ 0 <= m \/ n <= 0 /\ m <= 0. Proof. -assert (R : forall n : Z, 0 == n <-> n == 0) by (intros; split; apply Zeq_sym). -intros n m. repeat rewrite Zlt_eq_cases. repeat rewrite R. -rewrite Zlt_0_mul, Zeq_mul_0. -pose proof (Zlt_trichotomy n 0); pose proof (Zlt_trichotomy m 0). tauto. +assert (R : forall n, 0 == n <-> n == 0) by (intros; split; apply eq_sym). +intros n m. repeat rewrite lt_eq_cases. repeat rewrite R. +rewrite lt_0_mul, eq_mul_0. +pose proof (lt_trichotomy n 0); pose proof (lt_trichotomy m 0). tauto. Qed. -Notation Zmul_nonneg := Zle_0_mul (only parsing). +Notation mul_nonneg := le_0_mul (only parsing). -Theorem Zle_mul_0 : - forall n m : Z, n * m <= 0 -> 0 <= n /\ m <= 0 \/ n <= 0 /\ 0 <= m. +Theorem le_mul_0 : + forall n m, n * m <= 0 -> 0 <= n /\ m <= 0 \/ n <= 0 /\ 0 <= m. Proof. -assert (R : forall n : Z, 0 == n <-> n == 0) by (intros; split; apply Zeq_sym). -intros n m. repeat rewrite Zlt_eq_cases. repeat rewrite R. -rewrite Zlt_mul_0, Zeq_mul_0. -pose proof (Zlt_trichotomy n 0); pose proof (Zlt_trichotomy m 0). tauto. +assert (R : forall n, 0 == n <-> n == 0) by (intros; split; apply eq_sym). +intros n m. repeat rewrite lt_eq_cases. repeat rewrite R. +rewrite lt_mul_0, eq_mul_0. +pose proof (lt_trichotomy n 0); pose proof (lt_trichotomy m 0). tauto. Qed. -Notation Zmul_nonpos := Zle_mul_0 (only parsing). +Notation mul_nonpos := le_mul_0 (only parsing). -Theorem Zle_0_square : forall n : Z, 0 <= n * n. +Theorem le_0_square : forall n, 0 <= n * n. Proof. -intro n; destruct (Zneg_nonneg_cases n). -apply Zlt_le_incl; now apply Zmul_neg_neg. -now apply Zmul_nonneg_nonneg. +intro n; destruct (neg_nonneg_cases n). +apply lt_le_incl; now apply mul_neg_neg. +now apply mul_nonneg_nonneg. Qed. -Notation Zsquare_nonneg := Zle_0_square (only parsing). +Notation square_nonneg := le_0_square (only parsing). -Theorem Znlt_square_0 : forall n : Z, ~ n * n < 0. +Theorem nlt_square_0 : forall n, ~ n * n < 0. Proof. -intros n H. apply -> Zlt_nge in H. apply H. apply Zsquare_nonneg. +intros n H. apply -> lt_nge in H. apply H. apply square_nonneg. Qed. -Theorem Zsquare_lt_mono_nonneg : forall n m : Z, 0 <= n -> n < m -> n * n < m * m. -Proof NZsquare_lt_mono_nonneg. - -Theorem Zsquare_lt_mono_nonpos : forall n m : Z, n <= 0 -> m < n -> n * n < m * m. +Theorem square_lt_mono_nonpos : forall n m, n <= 0 -> m < n -> n * n < m * m. Proof. -intros n m H1 H2. now apply Zmul_lt_mono_nonpos. +intros n m H1 H2. now apply mul_lt_mono_nonpos. Qed. -Theorem Zsquare_le_mono_nonneg : forall n m : Z, 0 <= n -> n <= m -> n * n <= m * m. -Proof NZsquare_le_mono_nonneg. - -Theorem Zsquare_le_mono_nonpos : forall n m : Z, n <= 0 -> m <= n -> n * n <= m * m. +Theorem square_le_mono_nonpos : forall n m, n <= 0 -> m <= n -> n * n <= m * m. Proof. -intros n m H1 H2. now apply Zmul_le_mono_nonpos. +intros n m H1 H2. now apply mul_le_mono_nonpos. Qed. -Theorem Zsquare_lt_simpl_nonneg : forall n m : Z, 0 <= m -> n * n < m * m -> n < m. -Proof NZsquare_lt_simpl_nonneg. - -Theorem Zsquare_le_simpl_nonneg : forall n m : Z, 0 <= m -> n * n <= m * m -> n <= m. -Proof NZsquare_le_simpl_nonneg. - -Theorem Zsquare_lt_simpl_nonpos : forall n m : Z, m <= 0 -> n * n < m * m -> m < n. +Theorem square_lt_simpl_nonpos : forall n m, m <= 0 -> n * n < m * m -> m < n. Proof. -intros n m H1 H2. destruct (Zle_gt_cases n 0). -destruct (NZlt_ge_cases m n). -assumption. assert (F : m * m <= n * n) by now apply Zsquare_le_mono_nonpos. -apply -> NZle_ngt in F. false_hyp H2 F. -now apply Zle_lt_trans with 0. +intros n m H1 H2. destruct (le_gt_cases n 0). +destruct (lt_ge_cases m n). +assumption. assert (F : m * m <= n * n) by now apply square_le_mono_nonpos. +apply -> le_ngt in F. false_hyp H2 F. +now apply le_lt_trans with 0. Qed. -Theorem Zsquare_le_simpl_nonpos : forall n m : NZ, m <= 0 -> n * n <= m * m -> m <= n. +Theorem square_le_simpl_nonpos : forall n m, m <= 0 -> n * n <= m * m -> m <= n. Proof. -intros n m H1 H2. destruct (NZle_gt_cases n 0). -destruct (NZle_gt_cases m n). -assumption. assert (F : m * m < n * n) by now apply Zsquare_lt_mono_nonpos. -apply -> NZlt_nge in F. false_hyp H2 F. -apply Zlt_le_incl; now apply NZle_lt_trans with 0. +intros n m H1 H2. destruct (le_gt_cases n 0). +destruct (le_gt_cases m n). +assumption. assert (F : m * m < n * n) by now apply square_lt_mono_nonpos. +apply -> lt_nge in F. false_hyp H2 F. +apply lt_le_incl; now apply le_lt_trans with 0. Qed. -Theorem Zmul_2_mono_l : forall n m : Z, n < m -> 1 + (1 + 1) * n < (1 + 1) * m. -Proof NZmul_2_mono_l. - -Theorem Zlt_1_mul_neg : forall n m : Z, n < -1 -> m < 0 -> 1 < n * m. +Theorem lt_1_mul_neg : forall n m, n < -1 -> m < 0 -> 1 < n * m. Proof. -intros n m H1 H2. apply -> (NZmul_lt_mono_neg_r m) in H1. -apply <- Zopp_pos_neg in H2. rewrite Zmul_opp_l, Zmul_1_l in H1. -now apply Zlt_1_l with (- m). +intros n m H1 H2. apply -> (mul_lt_mono_neg_r m) in H1. +apply <- opp_pos_neg in H2. rewrite mul_opp_l, mul_1_l in H1. +now apply lt_1_l with (- m). assumption. Qed. -Theorem Zlt_mul_n1_neg : forall n m : Z, 1 < n -> m < 0 -> n * m < -1. +Theorem lt_mul_n1_neg : forall n m, 1 < n -> m < 0 -> n * m < -1. Proof. -intros n m H1 H2. apply -> (NZmul_lt_mono_neg_r m) in H1. -rewrite Zmul_1_l in H1. now apply Zlt_n1_r with m. +intros n m H1 H2. apply -> (mul_lt_mono_neg_r m) in H1. +rewrite mul_1_l in H1. now apply lt_n1_r with m. assumption. Qed. -Theorem Zlt_mul_n1_pos : forall n m : Z, n < -1 -> 0 < m -> n * m < -1. +Theorem lt_mul_n1_pos : forall n m, n < -1 -> 0 < m -> n * m < -1. Proof. -intros n m H1 H2. apply -> (NZmul_lt_mono_pos_r m) in H1. -rewrite Zmul_opp_l, Zmul_1_l in H1. -apply <- Zopp_neg_pos in H2. now apply Zlt_n1_r with (- m). +intros n m H1 H2. apply -> (mul_lt_mono_pos_r m) in H1. +rewrite mul_opp_l, mul_1_l in H1. +apply <- opp_neg_pos in H2. now apply lt_n1_r with (- m). assumption. Qed. -Theorem Zlt_1_mul_l : forall n m : Z, 1 < n -> n * m < -1 \/ n * m == 0 \/ 1 < n * m. +Theorem lt_1_mul_l : forall n m, 1 < n -> + n * m < -1 \/ n * m == 0 \/ 1 < n * m. Proof. -intros n m H; destruct (Zlt_trichotomy m 0) as [H1 | [H1 | H1]]. -left. now apply Zlt_mul_n1_neg. -right; left; now rewrite H1, Zmul_0_r. -right; right; now apply Zlt_1_mul_pos. +intros n m H; destruct (lt_trichotomy m 0) as [H1 | [H1 | H1]]. +left. now apply lt_mul_n1_neg. +right; left; now rewrite H1, mul_0_r. +right; right; now apply lt_1_mul_pos. Qed. -Theorem Zlt_n1_mul_r : forall n m : Z, n < -1 -> n * m < -1 \/ n * m == 0 \/ 1 < n * m. +Theorem lt_n1_mul_r : forall n m, n < -1 -> + n * m < -1 \/ n * m == 0 \/ 1 < n * m. Proof. -intros n m H; destruct (Zlt_trichotomy m 0) as [H1 | [H1 | H1]]. -right; right. now apply Zlt_1_mul_neg. -right; left; now rewrite H1, Zmul_0_r. -left. now apply Zlt_mul_n1_pos. +intros n m H; destruct (lt_trichotomy m 0) as [H1 | [H1 | H1]]. +right; right. now apply lt_1_mul_neg. +right; left; now rewrite H1, mul_0_r. +left. now apply lt_mul_n1_pos. Qed. -Theorem Zeq_mul_1 : forall n m : Z, n * m == 1 -> n == 1 \/ n == -1. +Theorem eq_mul_1 : forall n m, n * m == 1 -> n == 1 \/ n == -1. Proof. assert (F : ~ 1 < -1). intro H. -assert (H1 : -1 < 0). apply <- Zopp_neg_pos. apply Zlt_succ_diag_r. -assert (H2 : 1 < 0) by now apply Zlt_trans with (-1). false_hyp H2 Znlt_succ_diag_l. -Z0_pos_neg n. -intros m H; rewrite Zmul_0_l in H; false_hyp H Zneq_succ_diag_r. -intros n H; split; apply <- Zle_succ_l in H; le_elim H. -intros m H1; apply (Zlt_1_mul_l n m) in H. +assert (H1 : -1 < 0). apply <- opp_neg_pos. apply lt_succ_diag_r. +assert (H2 : 1 < 0) by now apply lt_trans with (-1). +false_hyp H2 nlt_succ_diag_l. +zero_pos_neg n. +intros m H; rewrite mul_0_l in H; false_hyp H neq_succ_diag_r. +intros n H; split; apply <- le_succ_l in H; le_elim H. +intros m H1; apply (lt_1_mul_l n m) in H. rewrite H1 in H; destruct H as [H | [H | H]]. -false_hyp H F. false_hyp H Zneq_succ_diag_l. false_hyp H Zlt_irrefl. +false_hyp H F. false_hyp H neq_succ_diag_l. false_hyp H lt_irrefl. intros; now left. -intros m H1; apply (Zlt_1_mul_l n m) in H. rewrite Zmul_opp_l in H1; -apply -> Zeq_opp_l in H1. rewrite H1 in H; destruct H as [H | [H | H]]. -false_hyp H Zlt_irrefl. apply -> Zeq_opp_l in H. rewrite Zopp_0 in H. -false_hyp H Zneq_succ_diag_l. false_hyp H F. -intros; right; symmetry; now apply Zopp_wd. +intros m H1; apply (lt_1_mul_l n m) in H. rewrite mul_opp_l in H1; +apply -> eq_opp_l in H1. rewrite H1 in H; destruct H as [H | [H | H]]. +false_hyp H lt_irrefl. apply -> eq_opp_l in H. rewrite opp_0 in H. +false_hyp H neq_succ_diag_l. false_hyp H F. +intros; right; symmetry; now apply opp_wd. Qed. -Theorem Zlt_mul_diag_l : forall n m : Z, n < 0 -> (1 < m <-> n * m < n). +Theorem lt_mul_diag_l : forall n m, n < 0 -> (1 < m <-> n * m < n). Proof. -intros n m H. stepr (n * m < n * 1) by now rewrite Zmul_1_r. -now apply Zmul_lt_mono_neg_l. +intros n m H. stepr (n * m < n * 1) by now rewrite mul_1_r. +now apply mul_lt_mono_neg_l. Qed. -Theorem Zlt_mul_diag_r : forall n m : Z, 0 < n -> (1 < m <-> n < n * m). +Theorem lt_mul_diag_r : forall n m, 0 < n -> (1 < m <-> n < n * m). Proof. -intros n m H. stepr (n * 1 < n * m) by now rewrite Zmul_1_r. -now apply Zmul_lt_mono_pos_l. +intros n m H. stepr (n * 1 < n * m) by now rewrite mul_1_r. +now apply mul_lt_mono_pos_l. Qed. -Theorem Zle_mul_diag_l : forall n m : Z, n < 0 -> (1 <= m <-> n * m <= n). +Theorem le_mul_diag_l : forall n m, n < 0 -> (1 <= m <-> n * m <= n). Proof. -intros n m H. stepr (n * m <= n * 1) by now rewrite Zmul_1_r. -now apply Zmul_le_mono_neg_l. +intros n m H. stepr (n * m <= n * 1) by now rewrite mul_1_r. +now apply mul_le_mono_neg_l. Qed. -Theorem Zle_mul_diag_r : forall n m : Z, 0 < n -> (1 <= m <-> n <= n * m). +Theorem le_mul_diag_r : forall n m, 0 < n -> (1 <= m <-> n <= n * m). Proof. -intros n m H. stepr (n * 1 <= n * m) by now rewrite Zmul_1_r. -now apply Zmul_le_mono_pos_l. +intros n m H. stepr (n * 1 <= n * m) by now rewrite mul_1_r. +now apply mul_le_mono_pos_l. Qed. -Theorem Zlt_mul_r : forall n m p : Z, 0 < n -> 1 < p -> n < m -> n < m * p. +Theorem lt_mul_r : forall n m p, 0 < n -> 1 < p -> n < m -> n < m * p. Proof. -intros. stepl (n * 1) by now rewrite Zmul_1_r. -apply Zmul_lt_mono_nonneg. -now apply Zlt_le_incl. assumption. apply Zle_0_1. assumption. +intros. stepl (n * 1) by now rewrite mul_1_r. +apply mul_lt_mono_nonneg. +now apply lt_le_incl. assumption. apply le_0_1. assumption. Qed. End ZMulOrderPropFunct. |