From cb24ec6fd2c79a317f98b7dad426ac3e9bbad56a Mon Sep 17 00:00:00 2001 From: emakarov Date: Fri, 16 Nov 2007 20:37:52 +0000 Subject: Added theorems; created NZPlusOrder from NTimesOrder. git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@10325 85f007b7-540e-0410-9357-904b9bb8a0f7 --- theories/Numbers/NatInt/NZPlusOrder.v | 166 ++++++++++++++++++++++++++++++++++ 1 file changed, 166 insertions(+) create mode 100644 theories/Numbers/NatInt/NZPlusOrder.v (limited to 'theories/Numbers/NatInt/NZPlusOrder.v') diff --git a/theories/Numbers/NatInt/NZPlusOrder.v b/theories/Numbers/NatInt/NZPlusOrder.v new file mode 100644 index 0000000000..9f1ba0f84b --- /dev/null +++ b/theories/Numbers/NatInt/NZPlusOrder.v @@ -0,0 +1,166 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* p + n < p + m. +Proof. +intros n m p; NZinduct p. +now do 2 rewrite NZplus_0_l. +intro p. do 2 rewrite NZplus_succ_l. now rewrite <- NZsucc_lt_mono. +Qed. + +Theorem NZplus_lt_mono_r : forall n m p : NZ, n < m <-> n + p < m + p. +Proof. +intros n m p. +rewrite (NZplus_comm n p); rewrite (NZplus_comm m p); apply NZplus_lt_mono_l. +Qed. + +Theorem NZplus_lt_mono : forall n m p q : NZ, n < m -> p < q -> n + p < m + q. +Proof. +intros n m p q H1 H2. +apply NZlt_trans with (m + p); +[now apply -> NZplus_lt_mono_r | now apply -> NZplus_lt_mono_l]. +Qed. + +Theorem NZplus_le_mono_l : forall n m p : NZ, n <= m <-> p + n <= p + m. +Proof. +intros n m p; NZinduct p. +now do 2 rewrite NZplus_0_l. +intro p. do 2 rewrite NZplus_succ_l. now rewrite <- NZsucc_le_mono. +Qed. + +Theorem NZplus_le_mono_r : forall n m p : NZ, n <= m <-> n + p <= m + p. +Proof. +intros n m p. +rewrite (NZplus_comm n p); rewrite (NZplus_comm m p); apply NZplus_le_mono_l. +Qed. + +Theorem NZplus_le_mono : forall n m p q : NZ, n <= m -> p <= q -> n + p <= m + q. +Proof. +intros n m p q H1 H2. +apply NZle_trans with (m + p); +[now apply -> NZplus_le_mono_r | now apply -> NZplus_le_mono_l]. +Qed. + +Theorem NZplus_lt_le_mono : forall n m p q : NZ, 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 -> NZplus_lt_mono_r | now apply -> NZplus_le_mono_l]. +Qed. + +Theorem NZplus_le_lt_mono : forall n m p q : NZ, 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 -> NZplus_le_mono_r | now apply -> NZplus_lt_mono_l]. +Qed. + +Theorem NZplus_pos_pos : forall n m : NZ, 0 < n -> 0 < m -> 0 < n + m. +Proof. +intros n m H1 H2. rewrite <- (NZplus_0_l 0). now apply NZplus_lt_mono. +Qed. + +Theorem NZplus_pos_nonneg : forall n m : NZ, 0 < n -> 0 <= m -> 0 < n + m. +Proof. +intros n m H1 H2. rewrite <- (NZplus_0_l 0). now apply NZplus_lt_le_mono. +Qed. + +Theorem NZplus_nonneg_pos : forall n m : NZ, 0 <= n -> 0 < m -> 0 < n + m. +Proof. +intros n m H1 H2. rewrite <- (NZplus_0_l 0). now apply NZplus_le_lt_mono. +Qed. + +Theorem NZplus_nonneg_nonneg : forall n m : NZ, 0 <= n -> 0 <= m -> 0 <= n + m. +Proof. +intros n m H1 H2. rewrite <- (NZplus_0_l 0). now apply NZplus_le_mono. +Qed. + +Theorem NZlt_plus_pos_l : forall n m : NZ, 0 < n -> m < n + m. +Proof. +intros n m H. apply -> (NZplus_lt_mono_r 0 n m) in H. +now rewrite NZplus_0_l in H. +Qed. + +Theorem NZlt_plus_pos_r : forall n m : NZ, 0 < n -> m < m + n. +Proof. +intros; rewrite NZplus_comm; now apply NZlt_plus_pos_l. +Qed. + +Theorem NZle_lt_plus_lt : forall n m p q : NZ, 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 (NZplus_le_mono q p n m H H1) as H3. apply <- NZnle_gt in H2. +false_hyp H3 H2. +Qed. + +Theorem NZlt_le_plus_lt : forall n m p q : NZ, 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 (NZplus_le_lt_mono q p n m H H1) as H3. apply <- NZnle_gt in H3. +false_hyp H2 H3. +Qed. + +Theorem NZle_le_plus_le : forall n m p q : NZ, 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 (NZplus_lt_le_mono q p n m H H1) as H3. apply <- NZnle_gt in H3. +false_hyp H2 H3. +Qed. + +Theorem NZplus_lt_cases : forall n m p q : NZ, 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 (NZplus_le_mono p n q m H1 H2) as H3. apply -> NZle_ngt in H3. +false_hyp H H3. +now right. now left. +Qed. + +Theorem NZplus_le_cases : forall n m p q : NZ, 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 NZplus_lt_mono. +apply -> NZle_ngt in H. false_hyp H3 H. +Qed. + +Theorem NZplus_neg_cases : forall n m : NZ, n + m < 0 -> n < 0 \/ m < 0. +Proof. +intros n m H; apply NZplus_lt_cases; now rewrite NZplus_0_l. +Qed. + +Theorem NZplus_pos_cases : forall n m : NZ, 0 < n + m -> 0 < n \/ 0 < m. +Proof. +intros n m H; apply NZplus_lt_cases; now rewrite NZplus_0_l. +Qed. + +Theorem NZplus_nonpos_cases : forall n m : NZ, n + m <= 0 -> n <= 0 \/ m <= 0. +Proof. +intros n m H; apply NZplus_le_cases; now rewrite NZplus_0_l. +Qed. + +Theorem NZplus_nonneg_cases : forall n m : NZ, 0 <= n + m -> 0 <= n \/ 0 <= m. +Proof. +intros n m H; apply NZplus_le_cases; now rewrite NZplus_0_l. +Qed. + +End NZPlusOrderPropFunct. + -- cgit v1.2.3