Integration of theories/Ints/Z/* in ZArith and large cleanup and extension of Zdiv

Some details: 

 - ZAux.v is the only file left in Ints/Z. The few elements that remain in it
   are rather specific or compatibility oriented. Others parts and files have 
   been either deleted when unused or pushed into some place of ZArith. 
 - Ints/List/ is removed since it was not needed anymore
 - Ints/Tactic.v disappear: some of its tactic were unused, some already in 
   Tactics.v (case_eq, f_equal instead of eq_tac), and the nice contradict 
   has been added to Tactics.v
 - Znumtheory inherits lots of results about Zdivide, rel_prime, prime, Zgcd, ...
 - A new file Zpow_facts inherits lots of results about Zpower. Placing them 
   into Zpower would have been difficult with respect to compatibility 
   (import of ring)
 - A few things added to Zmax, Zabs, Znat, Zsqrt, Zeven, Zorder
 - Adequate adaptations to Ints/num/* (mainly renaming of lemmas) 
 
Now, concerning Zdiv, the behavior when dividing by a negative number is now 
fully proved. When this was possible, existing lemmas has been extended, 
either from strictly positive to non-zero divisor, or even to arbitrary 
divisor (especially when playing with Zmod). These extended lemmas are named
with the suffix _full, whereas the original restrictive lemmas are retained 
for compatibility. Several lemmas now have shorter proofs (based on unicity 
lemmas). Lemmas are now more or less organized by themes (division and order, 
division and usual operations, etc). Three possible choices of spec for 
divisions on negative numbers are presented: this Zdiv, the ocaml approach
and the remainder-always-positive approach. The ugly behavior of Zopp 
with the current choice of Zdiv/Zmod is now fully covered. A embryo of 
division "a la Ocaml" is given: Odiv and Omod. 




git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@10291 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 27 insertions, 1 deletions

diff --git a/theories/ZArith/Zorder.v b/theories/ZArith/Zorder.v
index 45369561d..454560b85 100644
--- a/theories/ZArith/Zorder.v
+++ b/theories/ZArith/Zorder.v
@@ -7,7 +7,7 @@
 (************************************************************************)
 (*i $Id$ i*)
 
-(** Binary Integers (Pierre Crégut (CNET, Lannion, France) *)
+(** Binary Integers (Pierre Crégut (CNET, Lannion, France) *)
 
 Require Import BinPos.
 Require Import BinInt.
@@ -997,5 +997,31 @@ Proof.
     rewrite <- Zplus_assoc; rewrite Zplus_opp_l; rewrite Zplus_0_r; exact H. 
 Qed.
 
+Lemma Zmult_lt_compat:
+  forall n m p q : Z, 0 <= n < p -> 0 <= m < q -> n * m < p * q.
+Proof.
+  intros n m p q (H1, H2) (H3,H4).
+  assert (0<p) by (apply Zle_lt_trans with n; auto).
+  assert (0<q) by (apply Zle_lt_trans with m; auto).
+  case Zle_lt_or_eq with (1 := H1); intros H5; auto with zarith.
+  case Zle_lt_or_eq with (1 := H3); intros H6; auto with zarith.
+  apply Zlt_trans with (n * q).
+  apply Zmult_lt_compat_l; auto.
+  apply Zmult_lt_compat_r; auto with zarith.
+  rewrite <- H6; rewrite Zmult_0_r; apply Zmult_lt_0_compat; auto with zarith.
+  rewrite <- H5; simpl; apply Zmult_lt_0_compat; auto with zarith.
+Qed.
+
+Lemma Zmult_lt_compat2: 
+  forall n m p q : Z, 0 < n <= p -> 0 < m < q -> n * m < p * q.
+Proof.
+  intros n m p q (H1, H2) (H3, H4).
+  apply Zle_lt_trans with (p * m).
+  apply Zmult_le_compat_r; auto.
+  apply Zlt_le_weak; auto.
+  apply Zmult_lt_compat_l; auto.
+  apply Zlt_le_trans with n; auto.
+Qed.
+
 (** For compatibility *)
 Notation Zlt_O_minus_lt := Zlt_0_minus_lt (only parsing).