diff options
Diffstat (limited to 'theories/Numbers/NatInt/NZMulOrder.v')
| -rw-r--r-- | theories/Numbers/NatInt/NZMulOrder.v | 325 |
1 files changed, 161 insertions, 164 deletions
diff --git a/theories/Numbers/NatInt/NZMulOrder.v b/theories/Numbers/NatInt/NZMulOrder.v index c707bf73..7b64a698 100644 --- a/theories/Numbers/NatInt/NZMulOrder.v +++ b/theories/Numbers/NatInt/NZMulOrder.v @@ -8,303 +8,300 @@ (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) -(*i $Id: NZMulOrder.v 11040 2008-06-03 00:04:16Z letouzey $ i*) +(*i $Id$ i*) Require Import NZAxioms. Require Import NZAddOrder. -Module NZMulOrderPropFunct (Import NZOrdAxiomsMod : NZOrdAxiomsSig). -Module Export NZAddOrderPropMod := NZAddOrderPropFunct NZOrdAxiomsMod. -Open Local Scope NatIntScope. +Module Type NZMulOrderPropSig (Import NZ : NZOrdAxiomsSig'). +Include NZAddOrderPropSig NZ. -Theorem NZmul_lt_pred : - forall p q n m : NZ, S p == q -> (p * n < p * m <-> q * n + m < q * m + n). +Theorem mul_lt_pred : + forall p q n m, S p == q -> (p * n < p * m <-> q * n + m < q * m + n). Proof. -intros p q n m H. rewrite <- H. do 2 rewrite NZmul_succ_l. -rewrite <- (NZadd_assoc (p * n) n m). -rewrite <- (NZadd_assoc (p * m) m n). -rewrite (NZadd_comm n m). now rewrite <- NZadd_lt_mono_r. +intros p q n m H. rewrite <- H. nzsimpl. +rewrite <- ! add_assoc, (add_comm n m). +now rewrite <- add_lt_mono_r. Qed. -Theorem NZmul_lt_mono_pos_l : forall p n m : NZ, 0 < p -> (n < m <-> p * n < p * m). +Theorem mul_lt_mono_pos_l : forall p n m, 0 < p -> (n < m <-> p * n < p * m). Proof. -NZord_induct p. -intros n m H; false_hyp H NZlt_irrefl. -intros p H IH n m H1. do 2 rewrite NZmul_succ_l. -le_elim H. assert (LR : forall n m : NZ, n < m -> p * n + n < p * m + m). -intros n1 m1 H2. apply NZadd_lt_mono; [now apply -> IH | assumption]. -split; [apply LR |]. intro H2. apply -> NZlt_dne; intro H3. -apply <- NZle_ngt in H3. le_elim H3. -apply NZlt_asymm in H2. apply H2. now apply LR. -rewrite H3 in H2; false_hyp H2 NZlt_irrefl. -rewrite <- H; do 2 rewrite NZmul_0_l; now do 2 rewrite NZadd_0_l. -intros p H1 _ n m H2. apply NZlt_asymm in H1. false_hyp H2 H1. +nzord_induct p. +intros n m H; false_hyp H lt_irrefl. +intros p H IH n m H1. nzsimpl. +le_elim H. assert (LR : forall n m, n < m -> p * n + n < p * m + m). +intros n1 m1 H2. apply add_lt_mono; [now apply -> IH | assumption]. +split; [apply LR |]. intro H2. apply -> lt_dne; intro H3. +apply <- le_ngt in H3. le_elim H3. +apply lt_asymm in H2. apply H2. now apply LR. +rewrite H3 in H2; false_hyp H2 lt_irrefl. +rewrite <- H; now nzsimpl. +intros p H1 _ n m H2. destruct (lt_asymm _ _ H1 H2). Qed. -Theorem NZmul_lt_mono_pos_r : forall p n m : NZ, 0 < p -> (n < m <-> n * p < m * p). +Theorem mul_lt_mono_pos_r : forall p n m, 0 < p -> (n < m <-> n * p < m * p). Proof. intros p n m. -rewrite (NZmul_comm n p); rewrite (NZmul_comm m p). now apply NZmul_lt_mono_pos_l. +rewrite (mul_comm n p), (mul_comm m p). now apply mul_lt_mono_pos_l. Qed. -Theorem NZmul_lt_mono_neg_l : forall p n m : NZ, p < 0 -> (n < m <-> p * m < p * n). +Theorem mul_lt_mono_neg_l : forall p n m, p < 0 -> (n < m <-> p * m < p * n). Proof. -NZord_induct p. -intros n m H; false_hyp H NZlt_irrefl. -intros p H1 _ n m H2. apply NZlt_succ_l in H2. apply <- NZnle_gt in H2. false_hyp H1 H2. -intros p H IH n m H1. apply <- NZle_succ_l in H. -le_elim H. assert (LR : forall n m : NZ, n < m -> p * m < p * n). -intros n1 m1 H2. apply (NZle_lt_add_lt n1 m1). -now apply NZlt_le_incl. do 2 rewrite <- NZmul_succ_l. now apply -> IH. -split; [apply LR |]. intro H2. apply -> NZlt_dne; intro H3. -apply <- NZle_ngt in H3. le_elim H3. -apply NZlt_asymm in H2. apply H2. now apply LR. -rewrite H3 in H2; false_hyp H2 NZlt_irrefl. -rewrite (NZmul_lt_pred p (S p)) by reflexivity. -rewrite H; do 2 rewrite NZmul_0_l; now do 2 rewrite NZadd_0_l. +nzord_induct p. +intros n m H; false_hyp H lt_irrefl. +intros p H1 _ n m H2. apply lt_succ_l in H2. apply <- nle_gt in H2. +false_hyp H1 H2. +intros p H IH n m H1. apply <- le_succ_l in H. +le_elim H. assert (LR : forall n m, n < m -> p * m < p * n). +intros n1 m1 H2. apply (le_lt_add_lt n1 m1). +now apply lt_le_incl. rewrite <- 2 mul_succ_l. now apply -> IH. +split; [apply LR |]. intro H2. apply -> lt_dne; intro H3. +apply <- le_ngt in H3. le_elim H3. +apply lt_asymm in H2. apply H2. now apply LR. +rewrite H3 in H2; false_hyp H2 lt_irrefl. +rewrite (mul_lt_pred p (S p)) by reflexivity. +rewrite H; do 2 rewrite mul_0_l; now do 2 rewrite add_0_l. Qed. -Theorem NZmul_lt_mono_neg_r : forall p n m : NZ, p < 0 -> (n < m <-> m * p < n * p). +Theorem mul_lt_mono_neg_r : forall p n m, p < 0 -> (n < m <-> m * p < n * p). Proof. intros p n m. -rewrite (NZmul_comm n p); rewrite (NZmul_comm m p). now apply NZmul_lt_mono_neg_l. +rewrite (mul_comm n p), (mul_comm m p). now apply mul_lt_mono_neg_l. Qed. -Theorem NZmul_le_mono_nonneg_l : forall n m p : NZ, 0 <= p -> n <= m -> p * n <= p * m. +Theorem mul_le_mono_nonneg_l : forall n m p, 0 <= p -> n <= m -> p * n <= p * m. Proof. intros n m p H1 H2. le_elim H1. -le_elim H2. apply NZlt_le_incl. now apply -> NZmul_lt_mono_pos_l. -apply NZeq_le_incl; now rewrite H2. -apply NZeq_le_incl; rewrite <- H1; now do 2 rewrite NZmul_0_l. +le_elim H2. apply lt_le_incl. now apply -> mul_lt_mono_pos_l. +apply eq_le_incl; now rewrite H2. +apply eq_le_incl; rewrite <- H1; now do 2 rewrite mul_0_l. Qed. -Theorem NZmul_le_mono_nonpos_l : forall n m p : NZ, p <= 0 -> n <= m -> p * m <= p * n. +Theorem mul_le_mono_nonpos_l : forall n m p, p <= 0 -> n <= m -> p * m <= p * n. Proof. intros n m p H1 H2. le_elim H1. -le_elim H2. apply NZlt_le_incl. now apply -> NZmul_lt_mono_neg_l. -apply NZeq_le_incl; now rewrite H2. -apply NZeq_le_incl; rewrite H1; now do 2 rewrite NZmul_0_l. +le_elim H2. apply lt_le_incl. now apply -> mul_lt_mono_neg_l. +apply eq_le_incl; now rewrite H2. +apply eq_le_incl; rewrite H1; now do 2 rewrite mul_0_l. Qed. -Theorem NZmul_le_mono_nonneg_r : forall n m p : NZ, 0 <= p -> n <= m -> n * p <= m * p. +Theorem mul_le_mono_nonneg_r : forall n m p, 0 <= p -> n <= m -> n * p <= m * p. Proof. -intros n m p H1 H2; rewrite (NZmul_comm n p); rewrite (NZmul_comm m p); -now apply NZmul_le_mono_nonneg_l. +intros n m p H1 H2; +rewrite (mul_comm n p), (mul_comm m p); now apply mul_le_mono_nonneg_l. Qed. -Theorem NZmul_le_mono_nonpos_r : forall n m p : NZ, p <= 0 -> n <= m -> m * p <= n * p. +Theorem mul_le_mono_nonpos_r : forall n m p, p <= 0 -> n <= m -> m * p <= n * p. Proof. -intros n m p H1 H2; rewrite (NZmul_comm n p); rewrite (NZmul_comm m p); -now apply NZmul_le_mono_nonpos_l. +intros n m p H1 H2; +rewrite (mul_comm n p), (mul_comm m p); now apply mul_le_mono_nonpos_l. Qed. -Theorem NZmul_cancel_l : forall n m p : NZ, p ~= 0 -> (p * n == p * m <-> n == m). +Theorem mul_cancel_l : forall n m p, p ~= 0 -> (p * n == p * m <-> n == m). Proof. intros n m p H; split; intro H1. -destruct (NZlt_trichotomy p 0) as [H2 | [H2 | H2]]. -apply -> NZeq_dne; intro H3. apply -> NZlt_gt_cases in H3. destruct H3 as [H3 | H3]. -assert (H4 : p * m < p * n); [now apply -> NZmul_lt_mono_neg_l |]. -rewrite H1 in H4; false_hyp H4 NZlt_irrefl. -assert (H4 : p * n < p * m); [now apply -> NZmul_lt_mono_neg_l |]. -rewrite H1 in H4; false_hyp H4 NZlt_irrefl. +destruct (lt_trichotomy p 0) as [H2 | [H2 | H2]]. +apply -> eq_dne; intro H3. apply -> lt_gt_cases in H3. destruct H3 as [H3 | H3]. +assert (H4 : p * m < p * n); [now apply -> mul_lt_mono_neg_l |]. +rewrite H1 in H4; false_hyp H4 lt_irrefl. +assert (H4 : p * n < p * m); [now apply -> mul_lt_mono_neg_l |]. +rewrite H1 in H4; false_hyp H4 lt_irrefl. false_hyp H2 H. -apply -> NZeq_dne; intro H3. apply -> NZlt_gt_cases in H3. destruct H3 as [H3 | H3]. -assert (H4 : p * n < p * m) by (now apply -> NZmul_lt_mono_pos_l). -rewrite H1 in H4; false_hyp H4 NZlt_irrefl. -assert (H4 : p * m < p * n) by (now apply -> NZmul_lt_mono_pos_l). -rewrite H1 in H4; false_hyp H4 NZlt_irrefl. +apply -> eq_dne; intro H3. apply -> lt_gt_cases in H3. destruct H3 as [H3 | H3]. +assert (H4 : p * n < p * m) by (now apply -> mul_lt_mono_pos_l). +rewrite H1 in H4; false_hyp H4 lt_irrefl. +assert (H4 : p * m < p * n) by (now apply -> mul_lt_mono_pos_l). +rewrite H1 in H4; false_hyp H4 lt_irrefl. now rewrite H1. Qed. -Theorem NZmul_cancel_r : forall n m p : NZ, p ~= 0 -> (n * p == m * p <-> n == m). +Theorem mul_cancel_r : forall n m p, p ~= 0 -> (n * p == m * p <-> n == m). Proof. -intros n m p. rewrite (NZmul_comm n p), (NZmul_comm m p); apply NZmul_cancel_l. +intros n m p. rewrite (mul_comm n p), (mul_comm m p); apply mul_cancel_l. Qed. -Theorem NZmul_id_l : forall n m : NZ, m ~= 0 -> (n * m == m <-> n == 1). +Theorem mul_id_l : forall n m, m ~= 0 -> (n * m == m <-> n == 1). Proof. intros n m H. -stepl (n * m == 1 * m) by now rewrite NZmul_1_l. now apply NZmul_cancel_r. +stepl (n * m == 1 * m) by now rewrite mul_1_l. now apply mul_cancel_r. Qed. -Theorem NZmul_id_r : forall n m : NZ, n ~= 0 -> (n * m == n <-> m == 1). +Theorem mul_id_r : forall n m, n ~= 0 -> (n * m == n <-> m == 1). Proof. -intros n m; rewrite NZmul_comm; apply NZmul_id_l. +intros n m; rewrite mul_comm; apply mul_id_l. Qed. -Theorem NZmul_le_mono_pos_l : forall n m p : NZ, 0 < p -> (n <= m <-> p * n <= p * m). +Theorem mul_le_mono_pos_l : forall n m p, 0 < p -> (n <= m <-> p * n <= p * m). Proof. -intros n m p H; do 2 rewrite NZlt_eq_cases. -rewrite (NZmul_lt_mono_pos_l p n m) by assumption. -now rewrite -> (NZmul_cancel_l n m p) by -(intro H1; rewrite H1 in H; false_hyp H NZlt_irrefl). +intros n m p H; do 2 rewrite lt_eq_cases. +rewrite (mul_lt_mono_pos_l p n m) by assumption. +now rewrite -> (mul_cancel_l n m p) by +(intro H1; rewrite H1 in H; false_hyp H lt_irrefl). Qed. -Theorem NZmul_le_mono_pos_r : forall n m p : NZ, 0 < p -> (n <= m <-> n * p <= m * p). +Theorem mul_le_mono_pos_r : forall n m p, 0 < p -> (n <= m <-> n * p <= m * p). Proof. -intros n m p. rewrite (NZmul_comm n p); rewrite (NZmul_comm m p); -apply NZmul_le_mono_pos_l. +intros n m p. rewrite (mul_comm n p), (mul_comm m p); apply mul_le_mono_pos_l. Qed. -Theorem NZmul_le_mono_neg_l : forall n m p : NZ, p < 0 -> (n <= m <-> p * m <= p * n). +Theorem mul_le_mono_neg_l : forall n m p, p < 0 -> (n <= m <-> p * m <= p * n). Proof. -intros n m p H; do 2 rewrite NZlt_eq_cases. -rewrite (NZmul_lt_mono_neg_l p n m); [| assumption]. -rewrite -> (NZmul_cancel_l m n p) by (intro H1; rewrite H1 in H; false_hyp H NZlt_irrefl). -now setoid_replace (n == m) with (m == n) using relation iff by (split; now intro). +intros n m p H; do 2 rewrite lt_eq_cases. +rewrite (mul_lt_mono_neg_l p n m); [| assumption]. +rewrite -> (mul_cancel_l m n p) + by (intro H1; rewrite H1 in H; false_hyp H lt_irrefl). +now setoid_replace (n == m) with (m == n) by (split; now intro). Qed. -Theorem NZmul_le_mono_neg_r : forall n m p : NZ, p < 0 -> (n <= m <-> m * p <= n * p). +Theorem mul_le_mono_neg_r : forall n m p, p < 0 -> (n <= m <-> m * p <= n * p). Proof. -intros n m p. rewrite (NZmul_comm n p); rewrite (NZmul_comm m p); -apply NZmul_le_mono_neg_l. +intros n m p. rewrite (mul_comm n p), (mul_comm m p); apply mul_le_mono_neg_l. Qed. -Theorem NZmul_lt_mono_nonneg : - forall n m p q : NZ, 0 <= n -> n < m -> 0 <= p -> p < q -> n * p < m * q. +Theorem mul_lt_mono_nonneg : + forall n m p q, 0 <= n -> n < m -> 0 <= p -> p < q -> n * p < m * q. Proof. intros n m p q H1 H2 H3 H4. -apply NZle_lt_trans with (m * p). -apply NZmul_le_mono_nonneg_r; [assumption | now apply NZlt_le_incl]. -apply -> NZmul_lt_mono_pos_l; [assumption | now apply NZle_lt_trans with n]. +apply le_lt_trans with (m * p). +apply mul_le_mono_nonneg_r; [assumption | now apply lt_le_incl]. +apply -> mul_lt_mono_pos_l; [assumption | now apply le_lt_trans with n]. Qed. (* There are still many variants of the theorem above. One can assume 0 < n or 0 < p or n <= m or p <= q. *) -Theorem NZmul_le_mono_nonneg : - forall n m p q : NZ, 0 <= n -> n <= m -> 0 <= p -> p <= q -> n * p <= m * q. +Theorem mul_le_mono_nonneg : + forall n m p q, 0 <= n -> n <= m -> 0 <= p -> p <= q -> n * p <= m * q. Proof. intros n m p q H1 H2 H3 H4. le_elim H2; le_elim H4. -apply NZlt_le_incl; now apply NZmul_lt_mono_nonneg. -rewrite <- H4; apply NZmul_le_mono_nonneg_r; [assumption | now apply NZlt_le_incl]. -rewrite <- H2; apply NZmul_le_mono_nonneg_l; [assumption | now apply NZlt_le_incl]. -rewrite H2; rewrite H4; now apply NZeq_le_incl. +apply lt_le_incl; now apply mul_lt_mono_nonneg. +rewrite <- H4; apply mul_le_mono_nonneg_r; [assumption | now apply lt_le_incl]. +rewrite <- H2; apply mul_le_mono_nonneg_l; [assumption | now apply lt_le_incl]. +rewrite H2; rewrite H4; now apply eq_le_incl. Qed. -Theorem NZmul_pos_pos : forall n m : NZ, 0 < n -> 0 < m -> 0 < n * m. +Theorem mul_pos_pos : forall n m, 0 < n -> 0 < m -> 0 < n * m. Proof. -intros n m H1 H2. -rewrite <- (NZmul_0_l m). now apply -> NZmul_lt_mono_pos_r. +intros n m H1 H2. rewrite <- (mul_0_l m). now apply -> mul_lt_mono_pos_r. Qed. -Theorem NZmul_neg_neg : forall n m : NZ, n < 0 -> m < 0 -> 0 < n * m. +Theorem mul_neg_neg : forall n m, n < 0 -> m < 0 -> 0 < n * m. Proof. -intros n m H1 H2. -rewrite <- (NZmul_0_l m). now apply -> NZmul_lt_mono_neg_r. +intros n m H1 H2. rewrite <- (mul_0_l m). now apply -> mul_lt_mono_neg_r. Qed. -Theorem NZmul_pos_neg : forall n m : NZ, 0 < n -> m < 0 -> n * m < 0. +Theorem mul_pos_neg : forall n m, 0 < n -> m < 0 -> n * m < 0. Proof. -intros n m H1 H2. -rewrite <- (NZmul_0_l m). now apply -> NZmul_lt_mono_neg_r. +intros n m H1 H2. rewrite <- (mul_0_l m). now apply -> mul_lt_mono_neg_r. Qed. -Theorem NZmul_neg_pos : forall n m : NZ, n < 0 -> 0 < m -> n * m < 0. +Theorem mul_neg_pos : forall n m, n < 0 -> 0 < m -> n * m < 0. Proof. -intros; rewrite NZmul_comm; now apply NZmul_pos_neg. +intros; rewrite mul_comm; now apply mul_pos_neg. Qed. -Theorem NZlt_1_mul_pos : forall n m : NZ, 1 < n -> 0 < m -> 1 < n * m. +Theorem mul_nonneg_nonneg : forall n m, 0 <= n -> 0 <= m -> 0 <= n*m. Proof. -intros n m H1 H2. apply -> (NZmul_lt_mono_pos_r m) in H1. -rewrite NZmul_1_l in H1. now apply NZlt_1_l with m. +intros. rewrite <- (mul_0_l m). apply mul_le_mono_nonneg; order. +Qed. + +Theorem lt_1_mul_pos : forall n m, 1 < n -> 0 < m -> 1 < n * m. +Proof. +intros n m H1 H2. apply -> (mul_lt_mono_pos_r m) in H1. +rewrite mul_1_l in H1. now apply lt_1_l with m. assumption. Qed. -Theorem NZeq_mul_0 : forall n m : NZ, n * m == 0 <-> n == 0 \/ m == 0. +Theorem eq_mul_0 : forall n m, n * m == 0 <-> n == 0 \/ m == 0. Proof. intros n m; split. -intro H; destruct (NZlt_trichotomy n 0) as [H1 | [H1 | H1]]; -destruct (NZlt_trichotomy m 0) as [H2 | [H2 | H2]]; +intro H; destruct (lt_trichotomy n 0) as [H1 | [H1 | H1]]; +destruct (lt_trichotomy m 0) as [H2 | [H2 | H2]]; try (now right); try (now left). -elimtype False; now apply (NZlt_neq 0 (n * m)); [apply NZmul_neg_neg |]. -elimtype False; now apply (NZlt_neq (n * m) 0); [apply NZmul_neg_pos |]. -elimtype False; now apply (NZlt_neq (n * m) 0); [apply NZmul_pos_neg |]. -elimtype False; now apply (NZlt_neq 0 (n * m)); [apply NZmul_pos_pos |]. -intros [H | H]. now rewrite H, NZmul_0_l. now rewrite H, NZmul_0_r. +exfalso; now apply (lt_neq 0 (n * m)); [apply mul_neg_neg |]. +exfalso; now apply (lt_neq (n * m) 0); [apply mul_neg_pos |]. +exfalso; now apply (lt_neq (n * m) 0); [apply mul_pos_neg |]. +exfalso; now apply (lt_neq 0 (n * m)); [apply mul_pos_pos |]. +intros [H | H]. now rewrite H, mul_0_l. now rewrite H, mul_0_r. Qed. -Theorem NZneq_mul_0 : forall n m : NZ, n ~= 0 /\ m ~= 0 <-> n * m ~= 0. +Theorem neq_mul_0 : forall n m, n ~= 0 /\ m ~= 0 <-> n * m ~= 0. Proof. intros n m; split; intro H. -intro H1; apply -> NZeq_mul_0 in H1. tauto. +intro H1; apply -> eq_mul_0 in H1. tauto. split; intro H1; rewrite H1 in H; -(rewrite NZmul_0_l in H || rewrite NZmul_0_r in H); now apply H. +(rewrite mul_0_l in H || rewrite mul_0_r in H); now apply H. Qed. -Theorem NZeq_square_0 : forall n : NZ, n * n == 0 <-> n == 0. +Theorem eq_square_0 : forall n, n * n == 0 <-> n == 0. Proof. -intro n; rewrite NZeq_mul_0; tauto. +intro n; rewrite eq_mul_0; tauto. Qed. -Theorem NZeq_mul_0_l : forall n m : NZ, n * m == 0 -> m ~= 0 -> n == 0. +Theorem eq_mul_0_l : forall n m, n * m == 0 -> m ~= 0 -> n == 0. Proof. -intros n m H1 H2. apply -> NZeq_mul_0 in H1. destruct H1 as [H1 | H1]. +intros n m H1 H2. apply -> eq_mul_0 in H1. destruct H1 as [H1 | H1]. assumption. false_hyp H1 H2. Qed. -Theorem NZeq_mul_0_r : forall n m : NZ, n * m == 0 -> n ~= 0 -> m == 0. +Theorem eq_mul_0_r : forall n m, n * m == 0 -> n ~= 0 -> m == 0. Proof. -intros n m H1 H2; apply -> NZeq_mul_0 in H1. destruct H1 as [H1 | H1]. +intros n m H1 H2; apply -> eq_mul_0 in H1. destruct H1 as [H1 | H1]. false_hyp H1 H2. assumption. Qed. -Theorem NZlt_0_mul : forall n m : NZ, 0 < n * m <-> (0 < n /\ 0 < m) \/ (m < 0 /\ n < 0). +Theorem lt_0_mul : forall n m, 0 < n * m <-> (0 < n /\ 0 < m) \/ (m < 0 /\ n < 0). Proof. intros n m; split; [intro H | intros [[H1 H2] | [H1 H2]]]. -destruct (NZlt_trichotomy n 0) as [H1 | [H1 | H1]]; -[| rewrite H1 in H; rewrite NZmul_0_l in H; false_hyp H NZlt_irrefl |]; -(destruct (NZlt_trichotomy m 0) as [H2 | [H2 | H2]]; -[| rewrite H2 in H; rewrite NZmul_0_r in H; false_hyp H NZlt_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 NZmul_neg_pos. -elimtype False; now apply (NZlt_asymm (n * m) 0). -assert (H3 : n * m < 0) by now apply NZmul_pos_neg. -elimtype False; now apply (NZlt_asymm (n * m) 0). -now apply NZmul_pos_pos. now apply NZmul_neg_neg. +assert (H3 : n * m < 0) by now apply mul_neg_pos. +exfalso; now apply (lt_asymm (n * m) 0). +assert (H3 : n * m < 0) by now apply mul_pos_neg. +exfalso; now apply (lt_asymm (n * m) 0). +now apply mul_pos_pos. now apply mul_neg_neg. Qed. -Theorem NZsquare_lt_mono_nonneg : forall n m : NZ, 0 <= n -> n < m -> n * n < m * m. +Theorem square_lt_mono_nonneg : forall n m, 0 <= n -> n < m -> n * n < m * m. Proof. -intros n m H1 H2. now apply NZmul_lt_mono_nonneg. +intros n m H1 H2. now apply mul_lt_mono_nonneg. Qed. -Theorem NZsquare_le_mono_nonneg : forall n m : NZ, 0 <= n -> n <= m -> n * n <= m * m. +Theorem square_le_mono_nonneg : forall n m, 0 <= n -> n <= m -> n * n <= m * m. Proof. -intros n m H1 H2. now apply NZmul_le_mono_nonneg. +intros n m H1 H2. now apply mul_le_mono_nonneg. Qed. (* The converse theorems require nonnegativity (or nonpositivity) of the other variable *) -Theorem NZsquare_lt_simpl_nonneg : forall n m : NZ, 0 <= m -> n * n < m * m -> n < m. +Theorem square_lt_simpl_nonneg : forall n m, 0 <= m -> n * n < m * m -> n < m. Proof. -intros n m H1 H2. destruct (NZlt_ge_cases n 0). -now apply NZlt_le_trans with 0. -destruct (NZlt_ge_cases n m). -assumption. assert (F : m * m <= n * n) by now apply NZsquare_le_mono_nonneg. -apply -> NZle_ngt in F. false_hyp H2 F. +intros n m H1 H2. destruct (lt_ge_cases n 0). +now apply lt_le_trans with 0. +destruct (lt_ge_cases n m). +assumption. assert (F : m * m <= n * n) by now apply square_le_mono_nonneg. +apply -> le_ngt in F. false_hyp H2 F. Qed. -Theorem NZsquare_le_simpl_nonneg : forall n m : NZ, 0 <= m -> n * n <= m * m -> n <= m. +Theorem square_le_simpl_nonneg : forall n m, 0 <= m -> n * n <= m * m -> n <= m. Proof. -intros n m H1 H2. destruct (NZlt_ge_cases n 0). -apply NZlt_le_incl; now apply NZlt_le_trans with 0. -destruct (NZle_gt_cases n m). -assumption. assert (F : m * m < n * n) by now apply NZsquare_lt_mono_nonneg. -apply -> NZlt_nge in F. false_hyp H2 F. +intros n m H1 H2. destruct (lt_ge_cases n 0). +apply lt_le_incl; now apply lt_le_trans with 0. +destruct (le_gt_cases n m). +assumption. assert (F : m * m < n * n) by now apply square_lt_mono_nonneg. +apply -> lt_nge in F. false_hyp H2 F. Qed. -Theorem NZmul_2_mono_l : forall n m : NZ, n < m -> 1 + (1 + 1) * n < (1 + 1) * m. +Theorem mul_2_mono_l : forall n m, n < m -> 1 + (1 + 1) * n < (1 + 1) * m. Proof. -intros n m H. apply <- NZle_succ_l in H. -apply -> (NZmul_le_mono_pos_l (S n) m (1 + 1)) in H. -repeat rewrite NZmul_add_distr_r in *; repeat rewrite NZmul_1_l in *. -repeat rewrite NZadd_succ_r in *. repeat rewrite NZadd_succ_l in *. rewrite NZadd_0_l. -now apply -> NZle_succ_l. -apply NZadd_pos_pos; now apply NZlt_succ_diag_r. +intros n m. rewrite <- le_succ_l, (mul_le_mono_pos_l (S n) m (1 + 1)). +rewrite !mul_add_distr_r; nzsimpl; now rewrite le_succ_l. +apply add_pos_pos; now apply lt_0_1. Qed. -End NZMulOrderPropFunct. +End NZMulOrderPropSig. |