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, 22 insertions, 16 deletions

diff --git a/theories/Numbers/Integer/Binary/ZBinary.v b/theories/Numbers/Integer/Binary/ZBinary.v
index 01d36854b1..f68aa0dbe4 100644
--- a/theories/Numbers/Integer/Binary/ZBinary.v
+++ b/theories/Numbers/Integer/Binary/ZBinary.v
@@ -10,7 +10,7 @@
 
 
 Require Import ZAxioms ZProperties BinInt Zcompare Zorder ZArith_dec
- Zbool Zeven Zsqrt_def Zpow_def Zlog_def Zgcd_def.
+ Zbool Zeven Zsqrt_def Zpow_def Zlog_def Zgcd_def Zdiv_def.
 
 Local Open Scope Z_scope.
 
@@ -170,6 +170,27 @@ Definition gcd_nonneg := Zgcd_nonneg.
 Definition divide := Zdivide'.
 Definition gcd := Zgcd.
 
+(** Div / Mod / Quot / Rem *)
+
+Program Instance div_wd : Proper (eq==>eq==>eq) Zdiv.
+Program Instance mod_wd : Proper (eq==>eq==>eq) Zmod.
+Program Instance quot_wd : Proper (eq==>eq==>eq) Zquot.
+Program Instance rem_wd : Proper (eq==>eq==>eq) Zrem.
+
+Definition div_mod := Z_div_mod_eq_full.
+Definition mod_pos_bound := Zmod_pos_bound.
+Definition mod_neg_bound := Zmod_neg_bound.
+Definition mod_bound_pos := fun a b (_:0<=a) => Zmod_pos_bound a b.
+Definition div := Zdiv.
+Definition modulo := Zmod.
+
+Definition quot_rem := fun a b (_:b<>0) => Z_quot_rem_eq a b.
+Definition rem_bound_pos := Zrem_bound.
+Definition rem_opp_l := fun a b (_:b<>0) => Zrem_opp_l a b.
+Definition rem_opp_r := fun a b (_:b<>0) => Zrem_opp_r a b.
+Definition quot := Zquot.
+Definition remainder := Zrem.
+
 (** We define [eq] only here to avoid refering to this [eq] above. *)
 
 Definition eq := (@eq Z).
@@ -212,18 +233,3 @@ exact Zadd_opp_r.
 Qed.
 
 Add Ring ZR : Zring.*)
-
-
-
-(*
-Theorem eq_equiv_e : forall x y : Z, E x y <-> e x y.
-Proof.
-intros x y; unfold E, e, Zeq_bool; split; intro H.
-rewrite H; now rewrite Zcompare_refl.
-rewrite eq_true_unfold_pos in H.
-assert (H1 : (x ?= y) = Eq).
-case_eq (x ?= y); intro H1; rewrite H1 in H; simpl in H;
-[reflexivity | discriminate H | discriminate H].
-now apply Zcompare_Eq_eq.
-Qed.
-*)