diff options
Diffstat (limited to 'theories/Numbers/NatInt/NZAddOrder.v')
| -rw-r--r-- | theories/Numbers/NatInt/NZAddOrder.v | 141 |
1 files changed, 64 insertions, 77 deletions
diff --git a/theories/Numbers/NatInt/NZAddOrder.v b/theories/Numbers/NatInt/NZAddOrder.v index 50d1c42f..97c12202 100644 --- a/theories/Numbers/NatInt/NZAddOrder.v +++ b/theories/Numbers/NatInt/NZAddOrder.v @@ -8,159 +8,146 @@ (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) -(*i $Id: NZAddOrder.v 11040 2008-06-03 00:04:16Z letouzey $ i*) +(*i $Id$ i*) -Require Import NZAxioms. -Require Import NZOrder. +Require Import NZAxioms NZBase NZMul NZOrder. -Module NZAddOrderPropFunct (Import NZOrdAxiomsMod : NZOrdAxiomsSig). -Module Export NZOrderPropMod := NZOrderPropFunct NZOrdAxiomsMod. -Open Local Scope NatIntScope. +Module Type NZAddOrderPropSig (Import NZ : NZOrdAxiomsSig'). +Include NZBasePropSig NZ <+ NZMulPropSig NZ <+ NZOrderPropSig NZ. -Theorem NZadd_lt_mono_l : forall n m p : NZ, n < m <-> p + n < p + m. +Theorem add_lt_mono_l : forall n m p, n < m <-> p + n < p + m. Proof. -intros n m p; NZinduct p. -now do 2 rewrite NZadd_0_l. -intro p. do 2 rewrite NZadd_succ_l. now rewrite <- NZsucc_lt_mono. +intros n m p; nzinduct p. now nzsimpl. +intro p. nzsimpl. now rewrite <- succ_lt_mono. Qed. -Theorem NZadd_lt_mono_r : forall n m p : NZ, n < m <-> n + p < m + p. +Theorem add_lt_mono_r : forall n m p, n < m <-> n + p < m + p. Proof. -intros n m p. -rewrite (NZadd_comm n p); rewrite (NZadd_comm m p); apply NZadd_lt_mono_l. +intros n m p. rewrite (add_comm n p), (add_comm m p); apply add_lt_mono_l. Qed. -Theorem NZadd_lt_mono : forall n m p q : NZ, n < m -> p < q -> n + p < m + q. +Theorem add_lt_mono : forall n m p q, n < m -> p < q -> n + p < m + q. Proof. intros n m p q H1 H2. -apply NZlt_trans with (m + p); -[now apply -> NZadd_lt_mono_r | now apply -> NZadd_lt_mono_l]. +apply lt_trans with (m + p); +[now apply -> add_lt_mono_r | now apply -> add_lt_mono_l]. Qed. -Theorem NZadd_le_mono_l : forall n m p : NZ, n <= m <-> p + n <= p + m. +Theorem add_le_mono_l : forall n m p, n <= m <-> p + n <= p + m. Proof. -intros n m p; NZinduct p. -now do 2 rewrite NZadd_0_l. -intro p. do 2 rewrite NZadd_succ_l. now rewrite <- NZsucc_le_mono. +intros n m p; nzinduct p. now nzsimpl. +intro p. nzsimpl. now rewrite <- succ_le_mono. Qed. -Theorem NZadd_le_mono_r : forall n m p : NZ, n <= m <-> n + p <= m + p. +Theorem add_le_mono_r : forall n m p, n <= m <-> n + p <= m + p. Proof. -intros n m p. -rewrite (NZadd_comm n p); rewrite (NZadd_comm m p); apply NZadd_le_mono_l. +intros n m p. rewrite (add_comm n p), (add_comm m p); apply add_le_mono_l. Qed. -Theorem NZadd_le_mono : forall n m p q : NZ, n <= m -> p <= q -> n + p <= m + q. +Theorem add_le_mono : forall n m p q, n <= m -> p <= q -> n + p <= m + q. Proof. intros n m p q H1 H2. -apply NZle_trans with (m + p); -[now apply -> NZadd_le_mono_r | now apply -> NZadd_le_mono_l]. +apply le_trans with (m + p); +[now apply -> add_le_mono_r | now apply -> add_le_mono_l]. Qed. -Theorem NZadd_lt_le_mono : forall n m p q : NZ, n < m -> p <= q -> n + p < m + q. +Theorem add_lt_le_mono : forall n m p q, n < m -> p <= q -> n + p < m + q. Proof. intros n m p q H1 H2. -apply NZlt_le_trans with (m + p); -[now apply -> NZadd_lt_mono_r | now apply -> NZadd_le_mono_l]. +apply lt_le_trans with (m + p); +[now apply -> add_lt_mono_r | now apply -> add_le_mono_l]. Qed. -Theorem NZadd_le_lt_mono : forall n m p q : NZ, n <= m -> p < q -> n + p < m + q. +Theorem add_le_lt_mono : forall n m p q, n <= m -> p < q -> n + p < m + q. Proof. intros n m p q H1 H2. -apply NZle_lt_trans with (m + p); -[now apply -> NZadd_le_mono_r | now apply -> NZadd_lt_mono_l]. +apply le_lt_trans with (m + p); +[now apply -> add_le_mono_r | now apply -> add_lt_mono_l]. Qed. -Theorem NZadd_pos_pos : forall n m : NZ, 0 < n -> 0 < m -> 0 < n + m. +Theorem add_pos_pos : forall n m, 0 < n -> 0 < m -> 0 < n + m. Proof. -intros n m H1 H2. rewrite <- (NZadd_0_l 0). now apply NZadd_lt_mono. +intros n m H1 H2. rewrite <- (add_0_l 0). now apply add_lt_mono. Qed. -Theorem NZadd_pos_nonneg : forall n m : NZ, 0 < n -> 0 <= m -> 0 < n + m. +Theorem add_pos_nonneg : forall n m, 0 < n -> 0 <= m -> 0 < n + m. Proof. -intros n m H1 H2. rewrite <- (NZadd_0_l 0). now apply NZadd_lt_le_mono. +intros n m H1 H2. rewrite <- (add_0_l 0). now apply add_lt_le_mono. Qed. -Theorem NZadd_nonneg_pos : forall n m : NZ, 0 <= n -> 0 < m -> 0 < n + m. +Theorem add_nonneg_pos : forall n m, 0 <= n -> 0 < m -> 0 < n + m. Proof. -intros n m H1 H2. rewrite <- (NZadd_0_l 0). now apply NZadd_le_lt_mono. +intros n m H1 H2. rewrite <- (add_0_l 0). now apply add_le_lt_mono. Qed. -Theorem NZadd_nonneg_nonneg : forall n m : NZ, 0 <= n -> 0 <= m -> 0 <= n + m. +Theorem add_nonneg_nonneg : forall n m, 0 <= n -> 0 <= m -> 0 <= n + m. Proof. -intros n m H1 H2. rewrite <- (NZadd_0_l 0). now apply NZadd_le_mono. +intros n m H1 H2. rewrite <- (add_0_l 0). now apply add_le_mono. Qed. -Theorem NZlt_add_pos_l : forall n m : NZ, 0 < n -> m < n + m. +Theorem lt_add_pos_l : forall n m, 0 < n -> m < n + m. Proof. -intros n m H. apply -> (NZadd_lt_mono_r 0 n m) in H. -now rewrite NZadd_0_l in H. +intros n m. rewrite (add_lt_mono_r 0 n m). now nzsimpl. Qed. -Theorem NZlt_add_pos_r : forall n m : NZ, 0 < n -> m < m + n. +Theorem lt_add_pos_r : forall n m, 0 < n -> m < m + n. Proof. -intros; rewrite NZadd_comm; now apply NZlt_add_pos_l. +intros; rewrite add_comm; now apply lt_add_pos_l. Qed. -Theorem NZle_lt_add_lt : forall n m p q : NZ, n <= m -> p + m < q + n -> p < q. +Theorem le_lt_add_lt : forall n m p q, n <= m -> p + m < q + n -> p < q. Proof. -intros n m p q H1 H2. destruct (NZle_gt_cases q p); [| assumption]. -pose proof (NZadd_le_mono q p n m H H1) as H3. apply <- NZnle_gt in H2. -false_hyp H3 H2. +intros n m p q H1 H2. destruct (le_gt_cases q p); [| assumption]. +contradict H2. rewrite nlt_ge. now apply add_le_mono. Qed. -Theorem NZlt_le_add_lt : forall n m p q : NZ, n < m -> p + m <= q + n -> p < q. +Theorem lt_le_add_lt : forall n m p q, n < m -> p + m <= q + n -> p < q. Proof. -intros n m p q H1 H2. destruct (NZle_gt_cases q p); [| assumption]. -pose proof (NZadd_le_lt_mono q p n m H H1) as H3. apply <- NZnle_gt in H3. -false_hyp H2 H3. +intros n m p q H1 H2. destruct (le_gt_cases q p); [| assumption]. +contradict H2. rewrite nle_gt. now apply add_le_lt_mono. Qed. -Theorem NZle_le_add_le : forall n m p q : NZ, n <= m -> p + m <= q + n -> p <= q. +Theorem le_le_add_le : forall n m p q, n <= m -> p + m <= q + n -> p <= q. Proof. -intros n m p q H1 H2. destruct (NZle_gt_cases p q); [assumption |]. -pose proof (NZadd_lt_le_mono q p n m H H1) as H3. apply <- NZnle_gt in H3. -false_hyp H2 H3. +intros n m p q H1 H2. destruct (le_gt_cases p q); [assumption |]. +contradict H2. rewrite nle_gt. now apply add_lt_le_mono. Qed. -Theorem NZadd_lt_cases : forall n m p q : NZ, n + m < p + q -> n < p \/ m < q. +Theorem add_lt_cases : forall n m p q, n + m < p + q -> n < p \/ m < q. Proof. intros n m p q H; -destruct (NZle_gt_cases p n) as [H1 | H1]. -destruct (NZle_gt_cases q m) as [H2 | H2]. -pose proof (NZadd_le_mono p n q m H1 H2) as H3. apply -> NZle_ngt in H3. -false_hyp H H3. -now right. now left. +destruct (le_gt_cases p n) as [H1 | H1]; [| now left]. +destruct (le_gt_cases q m) as [H2 | H2]; [| now right]. +contradict H; rewrite nlt_ge. now apply add_le_mono. Qed. -Theorem NZadd_le_cases : forall n m p q : NZ, n + m <= p + q -> n <= p \/ m <= q. +Theorem add_le_cases : forall n m p q, n + m <= p + q -> n <= p \/ m <= q. Proof. intros n m p q H. -destruct (NZle_gt_cases n p) as [H1 | H1]. now left. -destruct (NZle_gt_cases m q) as [H2 | H2]. now right. -assert (H3 : p + q < n + m) by now apply NZadd_lt_mono. -apply -> NZle_ngt in H. false_hyp H3 H. +destruct (le_gt_cases n p) as [H1 | H1]. now left. +destruct (le_gt_cases m q) as [H2 | H2]. now right. +contradict H; rewrite nle_gt. now apply add_lt_mono. Qed. -Theorem NZadd_neg_cases : forall n m : NZ, n + m < 0 -> n < 0 \/ m < 0. +Theorem add_neg_cases : forall n m, n + m < 0 -> n < 0 \/ m < 0. Proof. -intros n m H; apply NZadd_lt_cases; now rewrite NZadd_0_l. +intros n m H; apply add_lt_cases; now nzsimpl. Qed. -Theorem NZadd_pos_cases : forall n m : NZ, 0 < n + m -> 0 < n \/ 0 < m. +Theorem add_pos_cases : forall n m, 0 < n + m -> 0 < n \/ 0 < m. Proof. -intros n m H; apply NZadd_lt_cases; now rewrite NZadd_0_l. +intros n m H; apply add_lt_cases; now nzsimpl. Qed. -Theorem NZadd_nonpos_cases : forall n m : NZ, n + m <= 0 -> n <= 0 \/ m <= 0. +Theorem add_nonpos_cases : forall n m, n + m <= 0 -> n <= 0 \/ m <= 0. Proof. -intros n m H; apply NZadd_le_cases; now rewrite NZadd_0_l. +intros n m H; apply add_le_cases; now nzsimpl. Qed. -Theorem NZadd_nonneg_cases : forall n m : NZ, 0 <= n + m -> 0 <= n \/ 0 <= m. +Theorem add_nonneg_cases : forall n m, 0 <= n + m -> 0 <= n \/ 0 <= m. Proof. -intros n m H; apply NZadd_le_cases; now rewrite NZadd_0_l. +intros n m H; apply add_le_cases; now nzsimpl. Qed. -End NZAddOrderPropFunct. +End NZAddOrderPropSig. |