1 files changed, 25 insertions, 76 deletions

diff --git a/theories/Numbers/Natural/Abstract/NMulOrder.v b/theories/Numbers/Natural/Abstract/NMulOrder.v
index aa21fb50..a2162b13 100644
--- a/theories/Numbers/Natural/Abstract/NMulOrder.v
+++ b/theories/Numbers/Natural/Abstract/NMulOrder.v
@@ -8,122 +8,71 @@
 (*                      Evgeny Makarov, INRIA, 2007                     *)
 (************************************************************************)
 
-(*i $Id: NMulOrder.v 11040 2008-06-03 00:04:16Z letouzey $ i*)
+(*i $Id$ i*)
 
 Require Export NAddOrder.
 
-Module NMulOrderPropFunct (Import NAxiomsMod : NAxiomsSig).
-Module Export NAddOrderPropMod := NAddOrderPropFunct NAxiomsMod.
-Open Local Scope NatScope.
+Module NMulOrderPropFunct (Import N : NAxiomsSig').
+Include NAddOrderPropFunct N.
 
-Theorem mul_lt_pred :
-  forall p q n m : N, S p == q -> (p * n < p * m <-> q * n + m < q * m + n).
-Proof NZmul_lt_pred.
+(** Theorems that are either not valid on Z or have different proofs
+    on N and Z *)
 
-Theorem mul_lt_mono_pos_l : forall p n m : N, 0 < p -> (n < m <-> p * n < p * m).
-Proof NZmul_lt_mono_pos_l.
-
-Theorem mul_lt_mono_pos_r : forall p n m : N, 0 < p -> (n < m <-> n * p < m * p).
-Proof NZmul_lt_mono_pos_r.
-
-Theorem mul_cancel_l : forall n m p : N, p ~= 0 -> (p * n == p * m <-> n == m).
-Proof NZmul_cancel_l.
-
-Theorem mul_cancel_r : forall n m p : N, p ~= 0 -> (n * p == m * p <-> n == m).
-Proof NZmul_cancel_r.
-
-Theorem mul_id_l : forall n m : N, m ~= 0 -> (n * m == m <-> n == 1).
-Proof NZmul_id_l.
-
-Theorem mul_id_r : forall n m : N, n ~= 0 -> (n * m == n <-> m == 1).
-Proof NZmul_id_r.
-
-Theorem mul_le_mono_pos_l : forall n m p : N, 0 < p -> (n <= m <-> p * n <= p * m).
-Proof NZmul_le_mono_pos_l.
-
-Theorem mul_le_mono_pos_r : forall n m p : N, 0 < p -> (n <= m <-> n * p <= m * p).
-Proof NZmul_le_mono_pos_r.
-
-Theorem mul_pos_pos : forall n m : N, 0 < n -> 0 < m -> 0 < n * m.
-Proof NZmul_pos_pos.
-
-Theorem lt_1_mul_pos : forall n m : N, 1 < n -> 0 < m -> 1 < n * m.
-Proof NZlt_1_mul_pos.
-
-Theorem eq_mul_0 : forall n m : N, n * m == 0 <-> n == 0 \/ m == 0.
-Proof NZeq_mul_0.
-
-Theorem neq_mul_0 : forall n m : N, n ~= 0 /\ m ~= 0 <-> n * m ~= 0.
-Proof NZneq_mul_0.
-
-Theorem eq_square_0 : forall n : N, n * n == 0 <-> n == 0.
-Proof NZeq_square_0.
-
-Theorem eq_mul_0_l : forall n m : N, n * m == 0 -> m ~= 0 -> n == 0.
-Proof NZeq_mul_0_l.
-
-Theorem eq_mul_0_r : forall n m : N, n * m == 0 -> n ~= 0 -> m == 0.
-Proof NZeq_mul_0_r.
-
-Theorem square_lt_mono : forall n m : N, n < m <-> n * n < m * m.
+Theorem square_lt_mono : forall n m, n < m <-> n * n < m * m.
 Proof.
 intros n m; split; intro;
-[apply NZsquare_lt_mono_nonneg | apply NZsquare_lt_simpl_nonneg];
+[apply square_lt_mono_nonneg | apply square_lt_simpl_nonneg];
 try assumption; apply le_0_l.
 Qed.
 
-Theorem square_le_mono : forall n m : N, n <= m <-> n * n <= m * m.
+Theorem square_le_mono : forall n m, n <= m <-> n * n <= m * m.
 Proof.
 intros n m; split; intro;
-[apply NZsquare_le_mono_nonneg | apply NZsquare_le_simpl_nonneg];
+[apply square_le_mono_nonneg | apply square_le_simpl_nonneg];
 try assumption; apply le_0_l.
 Qed.
 
-Theorem mul_2_mono_l : forall n m : N, n < m -> 1 + (1 + 1) * n < (1 + 1) * m.
-Proof NZmul_2_mono_l.
-
-(* Theorems that are either not valid on Z or have different proofs on N and Z *)
-
-Theorem mul_le_mono_l : forall n m p : N, n <= m -> p * n <= p * m.
+Theorem mul_le_mono_l : forall n m p, n <= m -> p * n <= p * m.
 Proof.
-intros; apply NZmul_le_mono_nonneg_l. apply le_0_l. assumption.
+intros; apply mul_le_mono_nonneg_l. apply le_0_l. assumption.
 Qed.
 
-Theorem mul_le_mono_r : forall n m p : N, n <= m -> n * p <= m * p.
+Theorem mul_le_mono_r : forall n m p, n <= m -> n * p <= m * p.
 Proof.
-intros; apply NZmul_le_mono_nonneg_r. apply le_0_l. assumption.
+intros; apply mul_le_mono_nonneg_r. apply le_0_l. assumption.
 Qed.
 
-Theorem mul_lt_mono : forall n m p q : N, n < m -> p < q -> n * p < m * q.
+Theorem mul_lt_mono : forall n m p q, n < m -> p < q -> n * p < m * q.
 Proof.
-intros; apply NZmul_lt_mono_nonneg; try assumption; apply le_0_l.
+intros; apply mul_lt_mono_nonneg; try assumption; apply le_0_l.
 Qed.
 
-Theorem mul_le_mono : forall n m p q : N, n <= m -> p <= q -> n * p <= m * q.
+Theorem mul_le_mono : forall n m p q, n <= m -> p <= q -> n * p <= m * q.
 Proof.
-intros; apply NZmul_le_mono_nonneg; try assumption; apply le_0_l.
+intros; apply mul_le_mono_nonneg; try assumption; apply le_0_l.
 Qed.
 
-Theorem lt_0_mul : forall n m : N, n * m > 0 <-> n > 0 /\ m > 0.
+Theorem lt_0_mul' : forall n m, n * m > 0 <-> n > 0 /\ m > 0.
 Proof.
 intros n m; split; [intro H | intros [H1 H2]].
-apply -> NZlt_0_mul in H. destruct H as [[H1 H2] | [H1 H2]]. now split. false_hyp H1 nlt_0_r.
-now apply NZmul_pos_pos.
+apply -> lt_0_mul in H. destruct H as [[H1 H2] | [H1 H2]]. now split.
+ false_hyp H1 nlt_0_r.
+now apply mul_pos_pos.
 Qed.
 
-Notation mul_pos := lt_0_mul (only parsing).
+Notation mul_pos := lt_0_mul' (only parsing).
 
-Theorem eq_mul_1 : forall n m : N, n * m == 1 <-> n == 1 /\ m == 1.
+Theorem eq_mul_1 : forall n m, n * m == 1 <-> n == 1 /\ m == 1.
 Proof.
 intros n m.
 split; [| intros [H1 H2]; now rewrite H1, H2, mul_1_l].
-intro H; destruct (NZlt_trichotomy n 1) as [H1 | [H1 | H1]].
+intro H; destruct (lt_trichotomy n 1) as [H1 | [H1 | H1]].
 apply -> lt_1_r in H1. rewrite H1, mul_0_l in H. false_hyp H neq_0_succ.
 rewrite H1, mul_1_l in H; now split.
 destruct (eq_0_gt_0_cases m) as [H2 | H2].
 rewrite H2, mul_0_r in H; false_hyp H neq_0_succ.
 apply -> (mul_lt_mono_pos_r m) in H1; [| assumption]. rewrite mul_1_l in H1.
-assert (H3 : 1 < n * m) by now apply (lt_1_l 0 m).
+assert (H3 : 1 < n * m) by now apply (lt_1_l m).
 rewrite H in H3; false_hyp H3 lt_irrefl.
 Qed.