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Diffstat (limited to 'theories/Numbers/Integer/Abstract/ZMulOrder.v')
| -rw-r--r-- | theories/Numbers/Integer/Abstract/ZMulOrder.v | 343 |
1 files changed, 343 insertions, 0 deletions
diff --git a/theories/Numbers/Integer/Abstract/ZMulOrder.v b/theories/Numbers/Integer/Abstract/ZMulOrder.v new file mode 100644 index 00000000..e3f1d9aa --- /dev/null +++ b/theories/Numbers/Integer/Abstract/ZMulOrder.v @@ -0,0 +1,343 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) +(* Evgeny Makarov, INRIA, 2007 *) +(************************************************************************) + +(*i $Id: ZMulOrder.v 11040 2008-06-03 00:04:16Z letouzey $ i*) + +Require Export ZAddOrder. + +Module ZMulOrderPropFunct (Import ZAxiomsMod : ZAxiomsSig). +Module Export ZAddOrderPropMod := ZAddOrderPropFunct ZAxiomsMod. +Open Local Scope IntScope. + +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. + +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. +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]. +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. +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. +Qed. + +Theorem Zmul_nonpos_nonpos : forall n m : Z, 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. +Qed. + +Theorem Zmul_nonneg_nonpos : forall n m : Z, 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. +Qed. + +Theorem Zmul_nonpos_nonneg : forall n m : Z, n <= 0 -> 0 <= m -> n * m <= 0. +Proof. +intros; rewrite Zmul_comm; now apply Zmul_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. + +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. +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 |]); +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. +Qed. + +Notation Zmul_neg := Zlt_mul_0 (only parsing). + +Theorem Zle_0_mul : + forall n m : Z, 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_symm). +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. +Qed. + +Notation Zmul_nonneg := Zle_0_mul (only parsing). + +Theorem Zle_mul_0 : + forall n m : Z, 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_symm). +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. +Qed. + +Notation Zmul_nonpos := Zle_mul_0 (only parsing). + +Theorem Zle_0_square : forall n : Z, 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. +Qed. + +Notation Zsquare_nonneg := Zle_0_square (only parsing). + +Theorem Znlt_square_0 : forall n : Z, ~ n * n < 0. +Proof. +intros n H. apply -> Zlt_nge in H. apply H. apply Zsquare_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. +Proof. +intros n m H1 H2. now apply Zmul_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. +Proof. +intros n m H1 H2. now apply Zmul_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. +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. +Qed. + +Theorem Zsquare_le_simpl_nonpos : forall n m : NZ, 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. +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. +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). +assumption. +Qed. + +Theorem Zlt_mul_n1_neg : forall n m : Z, 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. +assumption. +Qed. + +Theorem Zlt_mul_n1_pos : forall n m : Z, 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). +assumption. +Qed. + +Theorem Zlt_1_mul_l : forall n m : Z, 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. +Qed. + +Theorem Zlt_n1_mul_r : forall n m : Z, 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. +Qed. + +Theorem Zeq_mul_1 : forall n m : Z, 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. +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. +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. +Qed. + +Theorem Zlt_mul_diag_l : forall n m : Z, 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. +Qed. + +Theorem Zlt_mul_diag_r : forall n m : Z, 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. +Qed. + +Theorem Zle_mul_diag_l : forall n m : Z, 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. +Qed. + +Theorem Zle_mul_diag_r : forall n m : Z, 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. +Qed. + +Theorem Zlt_mul_r : forall n m p : Z, 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. +Qed. + +End ZMulOrderPropFunct. + |