3 files changed, 49 insertions, 68 deletions

diff --git a/theories/Numbers/NatInt/NZOrder.v b/theories/Numbers/NatInt/NZOrder.v
index 5c6369fe49..cb3dd3093f 100644
--- a/theories/Numbers/NatInt/NZOrder.v
+++ b/theories/Numbers/NatInt/NZOrder.v
@@ -44,6 +44,13 @@ Proof.
 intros n m H1 H2; rewrite H2 in H1; false_hyp H1 NZlt_irrefl.
 Qed.
 
+Theorem NZlt_le_neq : forall n m : NZ, n < m <-> n <= m /\ n ~= m.
+Proof.
+intros n m; split; [intro H | intros [H1 H2]].
+split. le_less. now apply NZlt_neq.
+le_elim H1. assumption. false_hyp H1 H2.
+Qed.
+
 Theorem NZle_refl : forall n : NZ, n <= n.
 Proof.
 intro; now le_equal.
diff --git a/theories/Numbers/NatInt/NZPlusOrder.v b/theories/Numbers/NatInt/NZPlusOrder.v
deleted file mode 100644
index 6368fa5578..0000000000
--- a/theories/Numbers/NatInt/NZPlusOrder.v
+++ /dev/null
@@ -1,67 +0,0 @@
-Require Export NZPlus.
-Require Export NZOrder.
-
-Module NZPlusOrderPropFunct
-  (Import NZPlusMod : NZPlusSig)
-  (Import NZOrderMod : NZOrderSig with Module NZBaseMod := NZPlusMod.NZBaseMod).
-
-Module Export NZPlusPropMod := NZPlusPropFunct NZPlusMod.
-Module Export NZOrderPropMod := NZOrderPropFunct NZOrderMod.
-Open Local Scope NatIntScope.
-
-Theorem NZplus_lt_mono_l : forall n m p, 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_lt_mono.
-Qed.
-
-Theorem NZplus_lt_mono_r : forall n m p, 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, 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, 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, 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, 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, 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, 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.
-
-End NZPlusOrderPropFunct.
-
diff --git a/theories/Numbers/NatInt/NZTimesOrder.v b/theories/Numbers/NatInt/NZTimesOrder.v
index 6f702067da..2f3cf678b9 100644
--- a/theories/Numbers/NatInt/NZTimesOrder.v
+++ b/theories/Numbers/NatInt/NZTimesOrder.v
@@ -61,6 +61,37 @@ 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].
@@ -75,7 +106,7 @@ 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_lt : forall n m p q : NZ, n <= m -> p + m <= q + n -> p <= q.
+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.
@@ -370,4 +401,14 @@ elimtype False; now apply (NZlt_asymm (n * m) 0).
 now apply NZtimes_neg_pos. now apply NZtimes_pos_neg.
 Qed.
 
+Theorem NZtimes_2_mono_l : forall n m : NZ, n < m -> 1 + (1 + 1) * n < (1 + 1) * m.
+Proof.
+intros n m H. apply -> NZlt_le_succ in H.
+apply -> (NZtimes_le_mono_pos_l (S n) m (1 + 1)) in H.
+repeat rewrite NZtimes_plus_distr_r in *; repeat rewrite NZtimes_1_l in *.
+repeat rewrite NZplus_succ_r in *. repeat rewrite NZplus_succ_l in *. rewrite NZplus_0_l.
+now apply <- NZlt_le_succ.
+apply NZplus_pos_pos; now apply NZlt_succ_r.
+Qed.
+
 End NZTimesOrderPropFunct.