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, 4 insertions, 5 deletions

diff --git a/theories/Ints/num/GenAdd.v b/theories/Ints/num/GenAdd.v
index 9d4c579020..77f5e23016 100644
--- a/theories/Ints/num/GenAdd.v
+++ b/theories/Ints/num/GenAdd.v
@@ -10,7 +10,6 @@ Set Implicit Arguments.
 
 Require Import ZArith.
 Require Import ZAux.
-Require Import ZDivModAux.
 Require Import Basic_type.
 Require Import GenBase.
 
@@ -265,11 +264,11 @@ Section GenAdd.
   Lemma spec_ww_succ : forall x, [[ww_succ x]] = ([[x]] + 1) mod wwB.
   Proof.
    destruct x as [ |xh xl];simpl.
-   rewrite spec_ww_1;rewrite Zmod_def_small;trivial.
+   rewrite spec_ww_1;rewrite Zmod_small;trivial.
    split;[intro;discriminate|apply wwB_pos].
    rewrite <- Zplus_assoc;generalize (spec_w_succ_c xl);
    destruct (w_succ_c xl) as[l|l];intro H;unfold interp_carry in H;rewrite <-H.
-   rewrite Zmod_def_small;trivial.
+   rewrite Zmod_small;trivial.
    rewrite wwB_wBwB;apply beta_mult;apply spec_to_Z.
    assert ([|l|] = 0). clear spec_ww_1 spec_w_1 spec_w_0.
     assert (H1:= spec_to_Z l); assert (H2:= spec_to_Z xl); omega.
@@ -281,10 +280,10 @@ Section GenAdd.
   Lemma spec_ww_add : forall x y, [[ww_add x y]] = ([[x]] + [[y]]) mod wwB.
   Proof.
    destruct x as [ |xh xl];intros y;simpl.
-   rewrite Zmod_def_small;trivial. apply spec_ww_to_Z;trivial.
+   rewrite Zmod_small;trivial. apply spec_ww_to_Z;trivial.
    destruct y as [ |yh yl].
    change [[W0]] with 0;rewrite Zplus_0_r.
-   rewrite Zmod_def_small;trivial. 
+   rewrite Zmod_small;trivial. 
     exact (spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW xh xl)).
    simpl. replace ([|xh|] * wB + [|xl|] + ([|yh|] * wB + [|yl|])) 
    with (([|xh|]+[|yh|])*wB + ([|xl|]+[|yl|])). 2:ring.