Integer division: quot and rem (trunc convention) in addition to div and mod

(floor convention).

We follow Haskell naming convention: quot and rem are for
Round-Toward-Zero (a.k.a Trunc, what Ocaml, C, Asm do by default, cf.
the ex-ZOdiv file), while div and mod are for Round-Toward-Bottom
(a.k.a Floor, what Coq does historically in Zdiv). We use unicode ÷
for quot, and infix rem for rem (which is actually remainder in
full). This way, both conventions can be used at the same time.

Definitions (and proofs of specifications) for div mod quot rem are
migrated in a new file Zdiv_def. Ex-ZOdiv file is now Zquot. With
this new organisation, no need for functor application in Zdiv and
Zquot.

On the abstract side, ZAxiomsSig now provides div mod quot rem.
Zproperties now contains properties of them. In NZDiv, we stop
splitting specifications in Common vs. Specific parts. Instead,
the NZ specification is be extended later, even if this leads to
a useless mod_bound_pos, subsumed by more precise axioms.

A few results in ZDivTrunc and ZDivFloor are improved (sgn stuff).

A few proofs in Nnat, Znat, Zabs are reworked (no more dependency
to Zmin, Zmax).

A lcm (least common multiple) is derived abstractly from gcd and
division (and hence available for nat N BigN Z BigZ :-).
In these new files NLcm and ZLcm, we also provide some combined
properties of div mod quot rem gcd.

We also provide a new file Zeuclid implementing a third division
convention, where the remainder is always positive. This file
instanciate the abstract one ZDivEucl. Operation names are
ZEuclid.div and ZEuclid.modulo.

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@13633 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 3 insertions, 3 deletions

diff --git a/theories/Numbers/Natural/Peano/NPeano.v b/theories/Numbers/Natural/Peano/NPeano.v
index ce8e03a5aa..60c59b3236 100644
--- a/theories/Numbers/Natural/Peano/NPeano.v
+++ b/theories/Numbers/Natural/Peano/NPeano.v
@@ -127,9 +127,9 @@ Proof.
  now rewrite <- mult_n_O, minus_diag, <- !plus_n_O in U.
 Qed.
 
-Lemma mod_upper_bound : forall x y, y<>0 -> x mod y < y.
+Lemma mod_bound_pos : forall x y, 0<=x -> 0<y -> 0 <= x mod y < y.
 Proof.
- intros x y Hy.
+ intros x y Hx Hy. split. auto with arith.
  destruct y; [ now elim Hy | clear Hy ].
  unfold modulo.
  apply le_n_S, le_minus.
@@ -515,7 +515,7 @@ Definition modulo := modulo.
 Program Instance div_wd : Proper (eq==>eq==>eq) div.
 Program Instance mod_wd : Proper (eq==>eq==>eq) modulo.
 Definition div_mod := div_mod.
-Definition mod_upper_bound := mod_upper_bound.
+Definition mod_bound_pos := mod_bound_pos.
 
 Definition divide := divide.
 Definition gcd := gcd.