A generic euclidean division in Numbers (Still Work-In-Progress)

  - For Z, we propose 3 conventions for the sign of the remainder...
  - Instanciation for nat in NPeano.
  - Beginning of instanciation in ZOdiv.

  Still many proofs to finish, etc, etc, but soon we will have a decent
  properties database for all divisions of all instances of Numbers (e.g. BigZ).

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12590 85f007b7-540e-0410-9357-904b9bb8a0f7

12 files changed, 2249 insertions, 134 deletions

diff --git a/theories/Numbers/Integer/Abstract/ZDivCoq.v b/theories/Numbers/Integer/Abstract/ZDivCoq.v
new file mode 100644
index 0000000000..402d520d5a
--- /dev/null
+++ b/theories/Numbers/Integer/Abstract/ZDivCoq.v
@@ -0,0 +1,390 @@
+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+
+(** Euclidean Division for integers
+
+    We use here the historical convention of Coq :
+    [a = bq+r /\ 0 < |r| < |b| /\ Sign(r) = Sgn(b)]
+ *)
+
+Require Import ZAxioms ZProperties NZDiv.
+
+Open Scope NumScope.
+
+Module Type ZDiv (Import Z : ZAxiomsSig).
+
+ Parameter Inline div : t -> t -> t.
+ Parameter Inline modulo : t -> t -> t.
+
+ Infix "/" := div : NumScope.
+ Infix "mod" := modulo (at level 40, no associativity) : NumScope.
+
+ Instance div_wd : Proper (eq==>eq==>eq) div.
+ Instance mod_wd : Proper (eq==>eq==>eq) modulo.
+
+ Axiom div_mod : forall a b, b ~= 0 -> a == b*(a/b) + (a mod b).
+ Axiom mod_pos_bound : forall a b, 0 < b -> 0 <= a mod b < b.
+ Axiom mod_neg_bound : forall a b, b < 0 -> b < a mod b <= 0.
+
+End ZDiv.
+
+Module Type ZDivSig := ZAxiomsSig <+ ZDiv.
+
+Module ZDivPropFunct (Import Z : ZDivSig).
+ (* TODO: en faire un arg du foncteur + comprendre le bug de SearchAbout *)
+ Module Import ZP := ZPropFunct Z.
+
+(** We benefit from what already exists for NZ *)
+
+ Module Z' <: NZDivSig.
+  Include Z.
+  Lemma mod_bound : forall a b, 0<=a -> 0<b -> 0 <= a mod b < b.
+  Proof. intros; apply mod_pos_bound; auto. Qed.
+ End Z'.
+ Module Import NZDivP := NZDivPropFunct Z'.
+
+(** Another formulation of the main equation *)
+
+Lemma mod_eq :
+ forall a b, b~=0 -> a mod b == a - b*(a/b).
+Proof.
+intros.
+rewrite <- add_move_l.
+symmetry. apply div_mod; auto.
+Qed.
+
+(** A few sign rules (simple ones) *)
+
+Lemma div_mod_opp_opp : forall a b, b~=0 ->
+ (-a/-b) == a/b /\ (-a) mod (-b) == -(a mod b).
+Proof.
+intros a b Hb.
+assert (-b ~= 0).
+ contradict Hb. rewrite eq_opp_l, opp_0 in Hb; auto.
+assert (EQ := opp_involutive a).
+rewrite (div_mod a b) in EQ at 2; auto.
+rewrite (div_mod (-a) (-b)) in EQ; auto.
+
+destruct (lt_ge_cases 0 b).
+rewrite opp_add_distr in EQ.
+rewrite <- mul_opp_l, opp_involutive in EQ.
+destruct (div_mod_unique b (-a/-b) (a/b) (-(-a mod -b)) (a mod b)); auto.
+rewrite <- (opp_involutive b) at 3.
+rewrite <- opp_lt_mono.
+rewrite opp_nonneg_nonpos.
+destruct (mod_neg_bound (-a) (-b)); auto.
+rewrite opp_neg_pos; auto.
+apply mod_pos_bound; auto.
+split; auto.
+rewrite eq_opp_r; auto.
+
+rewrite eq_opp_l in EQ.
+rewrite opp_add_distr in EQ.
+rewrite <- mul_opp_l in EQ.
+destruct (div_mod_unique (-b) (-a/-b) (a/b) (-a mod -b) (-(a mod b))); auto.
+apply mod_pos_bound; auto.
+rewrite opp_pos_neg; order.
+rewrite <- opp_lt_mono.
+rewrite opp_nonneg_nonpos.
+destruct (mod_neg_bound a b); intuition; order.
+Qed.
+
+Lemma div_opp_opp : forall a b, b~=0 -> -a/-b == a/b.
+Proof.
+intros; destruct (div_mod_opp_opp a b); auto.
+Qed.
+
+Lemma mod_opp_opp : forall a b, b~=0 -> (-a) mod (-b) == - (a mod b).
+Proof.
+intros; destruct (div_mod_opp_opp a b); auto.
+Qed.
+
+
+(** Uniqueness theorems *)
+
+
+Theorem div_mod_unique : forall b q1 q2 r1 r2 : t,
+  (0<=r1<b \/ b<r1<=0) -> (0<=r2<b \/ b<r2<=0) ->
+  b*q1+r1 == b*q2+r2 -> q1 == q2 /\ r1 == r2.
+Proof.
+intros b q1 q2 r1 r2 Hr1 Hr2 EQ.
+destruct Hr1; destruct Hr2; try (intuition; order).
+apply div_mod_unique with b; auto.
+rewrite <- opp_inj_wd in EQ.
+rewrite 2 opp_add_distr in EQ. rewrite <- 2 mul_opp_l in EQ.
+rewrite <- (opp_inj_wd r1 r2).
+apply div_mod_unique with (-b); auto.
+rewrite <- opp_lt_mono, opp_nonneg_nonpos; intuition.
+rewrite <- opp_lt_mono, opp_nonneg_nonpos; intuition.
+Qed.
+
+Theorem div_unique:
+ forall a b q r, (0<=r<b \/ b<r<=0) -> a == b*q + r -> q == a/b.
+Proof.
+intros a b q r Hr EQ.
+assert (Hb : b~=0) by (destruct Hr; intuition; order).
+rewrite (div_mod a b Hb) in EQ.
+destruct (div_mod_unique b (a/b) q (a mod b) r); auto.
+destruct Hr; [left; apply mod_pos_bound|right; apply mod_neg_bound];
+ intuition order.
+Qed.
+
+Theorem div_unique_pos:
+ forall a b q r, 0<=r<b -> a == b*q + r -> q == a/b.
+Proof. intros; apply div_unique with r; auto. Qed.
+
+Theorem div_unique_neg:
+ forall a b q r, 0<=r<b -> a == b*q + r -> q == a/b.
+Proof. intros; apply div_unique with r; auto. Qed.
+
+Theorem mod_unique:
+ forall a b q r, (0<=r<b \/ b<r<=0) -> a == b*q + r -> r == a mod b.
+Proof.
+intros a b q r Hr EQ.
+assert (Hb : b~=0) by (destruct Hr; intuition; order).
+rewrite (div_mod a b Hb) in EQ.
+destruct (div_mod_unique b (a/b) q (a mod b) r); auto.
+destruct Hr; [left; apply mod_pos_bound|right; apply mod_neg_bound];
+ intuition order.
+Qed.
+
+Theorem mod_unique_pos:
+ forall a b q r, 0<=r<b -> a == b*q + r -> r == a mod b.
+Proof. intros; apply mod_unique with q; auto. Qed.
+
+Theorem mod_unique_neg:
+ forall a b q r, b<r<=0 -> a == b*q + r -> r == a mod b.
+Proof. intros; apply mod_unique with q; auto. Qed.
+
+
+(** A division by itself returns 1 *)
+
+Ltac pos_or_neg a :=
+ let LT := fresh "LT" in
+ let LE := fresh "LE" in 
+ destruct (le_gt_cases 0 a) as [LE|LT]; [|rewrite <- opp_pos_neg in LT].
+
+Lemma div_same : forall a, a~=0 -> a/a == 1.
+Proof.
+intros. pos_or_neg a. apply div_same; order.
+rewrite <- div_opp_opp; auto. apply div_same; auto.
+Qed.
+
+Lemma mod_same : forall a, a~=0 -> a mod a == 0.
+Proof.
+intros. rewrite mod_eq, div_same; auto. nzsimpl. apply sub_diag.
+Qed.
+
+(** A division of a small number by a bigger one yields zero. *)
+
+Theorem div_small: forall a b, 0<=a<b -> a/b == 0.
+Proof. exact div_small. Qed.
+
+(** Same situation, in term of modulo: *)
+
+Theorem mod_small: forall a b, 0<=a<b -> a mod b == a.
+Proof. exact mod_small. Qed.
+
+(** * Basic values of divisions and modulo. *)
+
+Lemma div_0_l: forall a, a~=0 -> 0/a == 0.
+Proof.
+intros. pos_or_neg a. apply div_0_l; order.
+rewrite <- div_opp_opp, opp_0; auto. apply div_0_l; auto.
+Qed.
+
+Lemma mod_0_l: forall a, a~=0 -> 0 mod a == 0.
+Proof.
+intros; rewrite mod_eq, div_0_l; nzsimpl; auto.
+Qed.
+
+Lemma div_1_r: forall a, a/1 == a.
+Proof.
+intros. symmetry. apply div_unique with 0. left. split; order || apply lt_0_1.
+nzsimpl; auto.
+Qed.
+
+Lemma mod_1_r: forall a, a mod 1 == 0.
+Proof.
+intros. rewrite mod_eq, div_1_r; nzsimpl; auto using sub_diag.
+intro EQ; symmetry in EQ; revert EQ; apply lt_neq; apply lt_0_1.
+Qed.
+
+Lemma div_1_l: forall a, 1<a -> 1/a == 0.
+Proof. exact div_1_l. Qed.
+
+Lemma mod_1_l: forall a, 1<a -> 1 mod a == 1.
+Proof. exact mod_1_l. Qed.
+
+Lemma div_mul : forall a b, b~=0 -> (a*b)/b == a.
+Proof.
+intros. symmetry. apply div_unique with 0.
+destruct (lt_ge_cases 0 b); [left|right]; split; order.
+nzsimpl; apply mul_comm.
+Qed.
+
+Lemma mod_mul : forall a b, b~=0 -> (a*b) mod b == 0.
+Proof.
+intros. rewrite mod_eq, div_mul; auto. rewrite mul_comm; apply sub_diag.
+Qed.
+
+(** * Order results about mod and div *)
+
+(** A modulo cannot grow beyond its starting point. *)
+
+Theorem mod_le: forall a b, 0<=a -> 0<b -> a mod b <= a.
+Proof. exact mod_le. Qed.
+
+Theorem div_pos : forall a b, 0<=a -> 0<b -> 0<= a/b.
+Proof. exact div_pos. Qed.
+
+Lemma div_str_pos : forall a b, 0<b<=a -> 0 < a/b.
+Proof. exact div_str_pos. Qed.
+
+(* A REVOIR APRES LA REGLE DES SIGNES
+Lemma div_small_iff : forall a b, b~=0 -> (a/b==0 <-> 0<=a<b \/ b<a<=0).
+intros. apply div_small_iff; auto'. Qed.
+
+Lemma mod_small_iff : forall a b, b~=0 -> (a mod b == a <-> a<b).
+Proof. intros. apply mod_small_iff; auto'. Qed.
+
+Lemma div_str_pos_iff : forall a b, b~=0 -> (0<a/b <-> b<=a).
+Proof. intros. apply div_str_pos_iff; auto'. Qed.
+*)
+
+(** As soon as the divisor is strictly greater than 1,
+    the division is strictly decreasing. *)
+
+Lemma div_lt : forall a b, 0<a -> 1<b -> a/b < a.
+Proof. exact div_lt. Qed.
+
+(* STILL TODO !!
+
+(** [le] is compatible with a positive division. *)
+
+Lemma div_le_mono : forall a b c, 0<c -> a<=b -> a/c <= b/c.
+Proof.
+intros. destruct (le_gt_cases 0 a).
+apply div_le; auto.
+destruct (lt_ge_cases 0 b).
+apply le_trans with 0.
+ admit. (* !!! *)
+apply div_pos; order.
+Admitted. (* !!! *)
+
+Lemma mul_div_le : forall a b, b~=0 -> b*(a/b) <= a.
+Proof. intros. apply mul_div_le; auto'. Qed.
+
+Lemma mul_succ_div_gt: forall a b, b~=0 -> a < b*(S (a/b)).
+Proof. intros; apply mul_succ_div_gt; auto'. Qed.
+
+(** The previous inequality is exact iff the modulo is zero. *)
+
+Lemma div_exact : forall a b, b~=0 -> (a == b*(a/b) <-> a mod b == 0).
+Proof. intros. apply div_exact; auto'. Qed.
+
+(** Some additionnal inequalities about div. *)
+
+Theorem div_lt_upper_bound:
+  forall a b q, b~=0 -> a < b*q -> a/b < q.
+Proof. intros. apply div_lt_upper_bound; auto'. Qed.
+
+Theorem div_le_upper_bound:
+  forall a b q, b~=0 -> a <= b*q -> a/b <= q.
+Proof. intros; apply div_le_upper_bound; auto'. Qed.
+
+Theorem div_le_lower_bound:
+  forall a b q, b~=0 -> b*q <= a -> q <= a/b.
+Proof. intros; apply div_le_lower_bound; auto'. Qed.
+
+(** A division respects opposite monotonicity for the divisor *)
+
+Lemma div_le_compat_l: forall p q r, 0<q<r -> p/r <= p/q.
+Proof. intros. apply div_le_compat_l. auto'. auto. Qed.
+
+(** * Relations between usual operations and mod and div *)
+
+Lemma mod_add : forall a b c, c~=0 ->
+ (a + b * c) mod c == a mod c.
+Proof. intros. apply mod_add; auto'. Qed.
+
+Lemma div_add : forall a b c, c~=0 ->
+ (a + b * c) / c == a / c + b.
+Proof. intros. apply div_add; auto'. Qed.
+
+Lemma div_add_l: forall a b c, b~=0 ->
+ (a * b + c) / b == a + c / b.
+Proof. intros. apply div_add_l; auto'. Qed.
+
+(** Cancellations. *)
+
+Lemma div_mul_cancel_r : forall a b c, b~=0 -> c~=0 ->
+ (a*c)/(b*c) == a/b.
+Proof. intros. apply div_mul_cancel_r; auto'. Qed.
+
+Lemma div_mul_cancel_l : forall a b c, b~=0 -> c~=0 ->
+ (c*a)/(c*b) == a/b.
+Proof. intros. apply div_mul_cancel_l; auto'. Qed.
+
+Lemma mul_mod_distr_l: forall a b c, b~=0 -> c~=0 ->
+  (c*a) mod (c*b) == c * (a mod b).
+Proof. intros. apply mul_mod_distr_l; auto'. Qed.
+
+Lemma mul_mod_distr_r: forall a b c, b~=0 -> c~=0 ->
+  (a*c) mod (b*c) == (a mod b) * c.
+Proof. intros. apply mul_mod_distr_r; auto'. Qed.
+
+(** Operations modulo. *)
+
+Theorem mod_mod: forall a n, n~=0 ->
+ (a mod n) mod n == a mod n.
+Proof. intros. apply mod_mod; auto'. Qed.
+
+Lemma mul_mod_idemp_l : forall a b n, n~=0 ->
+ ((a mod n)*b) mod n == (a*b) mod n.
+Proof. intros. apply mul_mod_idemp_l; auto'. Qed.
+
+Lemma mul_mod_idemp_r : forall a b n, n~=0 ->
+ (a*(b mod n)) mod n == (a*b) mod n.
+Proof. intros. apply mul_mod_idemp_r; auto'. Qed.
+
+Theorem mul_mod: forall a b n, n~=0 ->
+ (a * b) mod n == ((a mod n) * (b mod n)) mod n.
+Proof. intros. apply mul_mod; auto'. Qed.
+
+Lemma add_mod_idemp_l : forall a b n, n~=0 ->
+ ((a mod n)+b) mod n == (a+b) mod n.
+Proof. intros. apply add_mod_idemp_l; auto'. Qed.
+
+Lemma add_mod_idemp_r : forall a b n, n~=0 ->
+ (a+(b mod n)) mod n == (a+b) mod n.
+Proof. intros. apply add_mod_idemp_r; auto'. Qed.
+
+Theorem add_mod: forall a b n, n~=0 ->
+ (a+b) mod n == (a mod n + b mod n) mod n.
+Proof. intros. apply add_mod; auto'. Qed.
+
+Lemma div_div : forall a b c, b~=0 -> c~=0 ->
+ (a/b)/c == a/(b*c).
+Proof. intros. apply div_div; auto'. Qed.
+
+(** A last inequality: *)
+
+Theorem div_mul_le:
+ forall a b c, b~=0 -> c*(a/b) <= (c*a)/b.
+Proof. intros. apply div_mul_le; auto'. Qed.
+
+(** mod is related to divisibility *)
+
+Lemma mod_divides : forall a b, b~=0 ->
+ (a mod b == 0 <-> exists c, a == b*c).
+Proof. intros. apply mod_divides; auto'. Qed.
+*)
+
+End ZDivPropFunct.
+
diff --git a/theories/Numbers/Integer/Abstract/ZDivMath.v b/theories/Numbers/Integer/Abstract/ZDivMath.v
new file mode 100644
index 0000000000..dfc9ee4bc7
--- /dev/null
+++ b/theories/Numbers/Integer/Abstract/ZDivMath.v
@@ -0,0 +1,396 @@
+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+
+(** Euclidean Division for integers
+
+    We use here the "mathematical" convention, i.e. Round-Toward-Bottom : 
+    [a = bq+r /\ 0 < r < |b| ]
+ *)
+
+Require Import ZAxioms ZProperties NZDiv.
+
+Open Scope NumScope.
+
+Module Type ZDiv (Import Z : ZAxiomsSig).
+
+ Parameter Inline div : t -> t -> t.
+ Parameter Inline modulo : t -> t -> t.
+
+ Infix "/" := div : NumScope.
+ Infix "mod" := modulo (at level 40, no associativity) : NumScope.
+
+ Instance div_wd : Proper (eq==>eq==>eq) div.
+ Instance mod_wd : Proper (eq==>eq==>eq) modulo.
+
+ Definition abs z := max z (-z).
+
+ Axiom div_mod : forall a b, b ~= 0 -> a == b*(a/b) + (a mod b).
+ Axiom mod_always_pos : forall a b, 0 <= a mod b < abs b.
+
+End ZDiv.
+
+Module Type ZDivSig := ZAxiomsSig <+ ZDiv.
+
+Module ZDivPropFunct (Import Z : ZDivSig).
+ (* TODO: en faire un arg du foncteur + comprendre le bug de SearchAbout *)
+ Module Import ZP := ZPropFunct Z.
+
+(** We benefit from what already exists for NZ *)
+
+ Module Z' <: NZDivSig.
+  Include Z.
+  Lemma mod_bound : forall a b, 0<=a -> 0<b -> 0 <= a mod b < b.
+  Proof.
+  intros. rewrite <- (max_l b (-b)) at 3.
+  apply mod_always_pos. 
+  apply le_trans with 0; [ rewrite opp_nonpos_nonneg |]; order.
+  Qed.
+ End Z'.
+ Module Import NZDivP := NZDivPropFunct Z'.
+
+(** Another formulation of the main equation *)
+
+Lemma mod_eq :
+ forall a b, b~=0 -> a mod b == a - b*(a/b).
+Proof.
+intros.
+rewrite <- add_move_l.
+symmetry. apply div_mod; auto.
+Qed.
+
+(* STILL TODO ...
+
+(** A few sign rules (simple ones) *)
+
+Lemma div_mod_opp_opp : forall a b, b~=0 ->
+ (-a/-b) == a/b /\ (-a) mod (-b) == -(a mod b).
+Proof.
+intros a b Hb.
+assert (-b ~= 0).
+ contradict Hb. rewrite eq_opp_l, opp_0 in Hb; auto.
+assert (EQ := opp_involutive a).
+rewrite (div_mod a b) in EQ at 2; auto.
+rewrite (div_mod (-a) (-b)) in EQ; auto.
+
+destruct (lt_ge_cases 0 b).
+rewrite opp_add_distr in EQ.
+rewrite <- mul_opp_l, opp_involutive in EQ.
+destruct (div_mod_unique b (-a/-b) (a/b) (-(-a mod -b)) (a mod b)); auto.
+rewrite <- (opp_involutive b) at 3.
+rewrite <- opp_lt_mono.
+rewrite opp_nonneg_nonpos.
+destruct (mod_neg_bound (-a) (-b)); auto.
+rewrite opp_neg_pos; auto.
+apply mod_pos_bound; auto.
+split; auto.
+rewrite eq_opp_r; auto.
+
+rewrite eq_opp_l in EQ.
+rewrite opp_add_distr in EQ.
+rewrite <- mul_opp_l in EQ.
+destruct (div_mod_unique (-b) (-a/-b) (a/b) (-a mod -b) (-(a mod b))); auto.
+apply mod_pos_bound; auto.
+rewrite opp_pos_neg; order.
+rewrite <- opp_lt_mono.
+rewrite opp_nonneg_nonpos.
+destruct (mod_neg_bound a b); intuition; order.
+Qed.
+
+Lemma div_opp_opp : forall a b, b~=0 -> -a/-b == a/b.
+Proof.
+intros; destruct (div_mod_opp_opp a b); auto.
+Qed.
+
+Lemma mod_opp_opp : forall a b, b~=0 -> (-a) mod (-b) == - (a mod b).
+Proof.
+intros; destruct (div_mod_opp_opp a b); auto.
+Qed.
+
+
+(** Uniqueness theorems *)
+
+
+Theorem div_mod_unique : forall b q1 q2 r1 r2 : t,
+  (0<=r1<b \/ b<r1<=0) -> (0<=r2<b \/ b<r2<=0) ->
+  b*q1+r1 == b*q2+r2 -> q1 == q2 /\ r1 == r2.
+Proof.
+intros b q1 q2 r1 r2 Hr1 Hr2 EQ.
+destruct Hr1; destruct Hr2; try (intuition; order).
+apply div_mod_unique with b; auto.
+rewrite <- opp_inj_wd in EQ.
+rewrite 2 opp_add_distr in EQ. rewrite <- 2 mul_opp_l in EQ.
+rewrite <- (opp_inj_wd r1 r2).
+apply div_mod_unique with (-b); auto.
+rewrite <- opp_lt_mono, opp_nonneg_nonpos; intuition.
+rewrite <- opp_lt_mono, opp_nonneg_nonpos; intuition.
+Qed.
+
+Theorem div_unique:
+ forall a b q r, (0<=r<b \/ b<r<=0) -> a == b*q + r -> q == a/b.
+Proof.
+intros a b q r Hr EQ.
+assert (Hb : b~=0) by (destruct Hr; intuition; order).
+rewrite (div_mod a b Hb) in EQ.
+destruct (div_mod_unique b (a/b) q (a mod b) r); auto.
+destruct Hr; [left; apply mod_pos_bound|right; apply mod_neg_bound];
+ intuition order.
+Qed.
+
+Theorem div_unique_pos:
+ forall a b q r, 0<=r<b -> a == b*q + r -> q == a/b.
+Proof. intros; apply div_unique with r; auto. Qed.
+
+Theorem div_unique_neg:
+ forall a b q r, 0<=r<b -> a == b*q + r -> q == a/b.
+Proof. intros; apply div_unique with r; auto. Qed.
+
+Theorem mod_unique:
+ forall a b q r, (0<=r<b \/ b<r<=0) -> a == b*q + r -> r == a mod b.
+Proof.
+intros a b q r Hr EQ.
+assert (Hb : b~=0) by (destruct Hr; intuition; order).
+rewrite (div_mod a b Hb) in EQ.
+destruct (div_mod_unique b (a/b) q (a mod b) r); auto.
+destruct Hr; [left; apply mod_pos_bound|right; apply mod_neg_bound];
+ intuition order.
+Qed.
+
+Theorem mod_unique_pos:
+ forall a b q r, 0<=r<b -> a == b*q + r -> r == a mod b.
+Proof. intros; apply mod_unique with q; auto. Qed.
+
+Theorem mod_unique_neg:
+ forall a b q r, b<r<=0 -> a == b*q + r -> r == a mod b.
+Proof. intros; apply mod_unique with q; auto. Qed.
+
+
+(** A division by itself returns 1 *)
+
+Ltac pos_or_neg a :=
+ destruct (le_gt_cases 0 a) as [LE|LT]; [|rewrite <- opp_pos_neg in LT].
+
+Lemma div_same : forall a, a~=0 -> a/a == 1.
+Proof.
+intros. pos_or_neg a. apply div_same; order.
+rewrite <- div_opp_opp; auto. apply div_same; auto.
+Qed.
+
+Lemma mod_same : forall a, a~=0 -> a mod a == 0.
+Proof.
+intros. rewrite mod_eq, div_same; auto. nzsimpl. apply sub_diag.
+Qed.
+
+(** A division of a small number by a bigger one yields zero. *)
+
+Theorem div_small: forall a b, 0<=a<b -> a/b == 0.
+Proof. exact div_small. Qed.
+
+(** Same situation, in term of modulo: *)
+
+Theorem mod_small: forall a b, 0<=a<b -> a mod b == a.
+Proof. exact mod_small. Qed.
+
+(** * Basic values of divisions and modulo. *)
+
+Lemma div_0_l: forall a, a~=0 -> 0/a == 0.
+Proof.
+intros. pos_or_neg a. apply div_0_l; order.
+rewrite <- div_opp_opp, opp_0; auto. apply div_0_l; auto.
+Qed.
+
+Lemma mod_0_l: forall a, a~=0 -> 0 mod a == 0.
+Proof.
+intros; rewrite mod_eq, div_0_l; nzsimpl; auto.
+Qed.
+
+Lemma div_1_r: forall a, a/1 == a.
+Proof.
+intros. symmetry. apply div_unique with 0. left. split; order || apply lt_0_1.
+nzsimpl; auto.
+Qed.
+
+Lemma mod_1_r: forall a, a mod 1 == 0.
+Proof.
+intros. rewrite mod_eq, div_1_r; nzsimpl; auto using sub_diag.
+intro EQ; symmetry in EQ; revert EQ; apply lt_neq; apply lt_0_1.
+Qed.
+
+Lemma div_1_l: forall a, 1<a -> 1/a == 0.
+Proof. exact div_1_l. Qed.
+
+Lemma mod_1_l: forall a, 1<a -> 1 mod a == 1.
+Proof. exact mod_1_l. Qed.
+
+Lemma div_mul : forall a b, b~=0 -> (a*b)/b == a.
+Proof.
+intros. symmetry. apply div_unique with 0.
+destruct (lt_ge_cases 0 b); [left|right]; split; order.
+nzsimpl; apply mul_comm.
+Qed.
+
+Lemma mod_mul : forall a b, b~=0 -> (a*b) mod b == 0.
+Proof.
+intros. rewrite mod_eq, div_mul; auto. rewrite mul_comm; apply sub_diag.
+Qed.
+
+(** * Order results about mod and div *)
+
+(** A modulo cannot grow beyond its starting point. *)
+
+Theorem mod_le: forall a b, 0<=a -> 0<b -> a mod b <= a.
+Proof. exact mod_le. Qed.
+
+Theorem div_pos : forall a b, 0<=a -> 0<b -> 0<= a/b.
+Proof. exact div_pos. Qed.
+
+Lemma div_str_pos : forall a b, 0<b<=a -> 0 < a/b.
+Proof. exact div_str_pos. Qed.
+
+(* A REVOIR APRES LA REGLE DES SIGNES
+Lemma div_small_iff : forall a b, b~=0 -> (a/b==0 <-> 0<=a<b \/ b<a<=0).
+intros. apply div_small_iff; auto'. Qed.
+
+Lemma mod_small_iff : forall a b, b~=0 -> (a mod b == a <-> a<b).
+Proof. intros. apply mod_small_iff; auto'. Qed.
+
+Lemma div_str_pos_iff : forall a b, b~=0 -> (0<a/b <-> b<=a).
+Proof. intros. apply div_str_pos_iff; auto'. Qed.
+*)
+
+(** As soon as the divisor is strictly greater than 1,
+    the division is strictly decreasing. *)
+
+Lemma div_lt : forall a b, 0<a -> 1<b -> a/b < a.
+Proof. exact div_lt. Qed.
+
+(* STILL TODO !!
+
+(** [le] is compatible with a positive division. *)
+
+Lemma div_le_mono: forall a b c, 0<c -> a<=b -> a/c <= b/c.
+Proof.
+intros. destruct (le_gt_cases 0 a).
+apply div_le_mono; auto.
+destruct (lt_ge_cases 0 b).
+apply le_trans with 0.
+ admit. (* !!! *)
+apply div_pos; order.
+Admitted. (* !!! *)
+
+Lemma mul_div_le : forall a b, b~=0 -> b*(a/b) <= a.
+Proof. intros. apply mul_div_le; auto'. Qed.
+
+Lemma mul_succ_div_gt: forall a b, b~=0 -> a < b*(S (a/b)).
+Proof. intros; apply mul_succ_div_gt; auto'. Qed.
+
+(** The previous inequality is exact iff the modulo is zero. *)
+
+Lemma div_exact : forall a b, b~=0 -> (a == b*(a/b) <-> a mod b == 0).
+Proof. intros. apply div_exact; auto'. Qed.
+
+(** Some additionnal inequalities about div. *)
+
+Theorem div_lt_upper_bound:
+  forall a b q, b~=0 -> a < b*q -> a/b < q.
+Proof. intros. apply div_lt_upper_bound; auto'. Qed.
+
+Theorem div_le_upper_bound:
+  forall a b q, b~=0 -> a <= b*q -> a/b <= q.
+Proof. intros; apply div_le_upper_bound; auto'. Qed.
+
+Theorem div_le_lower_bound:
+  forall a b q, b~=0 -> b*q <= a -> q <= a/b.
+Proof. intros; apply div_le_lower_bound; auto'. Qed.
+
+(** A division respects opposite monotonicity for the divisor *)
+
+Lemma div_le_compat_l: forall p q r, 0<q<r -> p/r <= p/q.
+Proof. intros. apply div_le_compat_l. auto'. auto. Qed.
+
+(** * Relations between usual operations and mod and div *)
+
+Lemma mod_add : forall a b c, c~=0 ->
+ (a + b * c) mod c == a mod c.
+Proof. intros. apply mod_add; auto'. Qed.
+
+Lemma div_add : forall a b c, c~=0 ->
+ (a + b * c) / c == a / c + b.
+Proof. intros. apply div_add; auto'. Qed.
+
+Lemma div_add_l: forall a b c, b~=0 ->
+ (a * b + c) / b == a + c / b.
+Proof. intros. apply div_add_l; auto'. Qed.
+
+(** Cancellations. *)
+
+Lemma div_mul_cancel_r : forall a b c, b~=0 -> c~=0 ->
+ (a*c)/(b*c) == a/b.
+Proof. intros. apply div_mul_cancel_r; auto'. Qed.
+
+Lemma div_mul_cancel_l : forall a b c, b~=0 -> c~=0 ->
+ (c*a)/(c*b) == a/b.
+Proof. intros. apply div_mul_cancel_l; auto'. Qed.
+
+Lemma mul_mod_distr_l: forall a b c, b~=0 -> c~=0 ->
+  (c*a) mod (c*b) == c * (a mod b).
+Proof. intros. apply mul_mod_distr_l; auto'. Qed.
+
+Lemma mul_mod_distr_r: forall a b c, b~=0 -> c~=0 ->
+  (a*c) mod (b*c) == (a mod b) * c.
+Proof. intros. apply mul_mod_distr_r; auto'. Qed.
+
+(** Operations modulo. *)
+
+Theorem mod_mod: forall a n, n~=0 ->
+ (a mod n) mod n == a mod n.
+Proof. intros. apply mod_mod; auto'. Qed.
+
+Lemma mul_mod_idemp_l : forall a b n, n~=0 ->
+ ((a mod n)*b) mod n == (a*b) mod n.
+Proof. intros. apply mul_mod_idemp_l; auto'. Qed.
+
+Lemma mul_mod_idemp_r : forall a b n, n~=0 ->
+ (a*(b mod n)) mod n == (a*b) mod n.
+Proof. intros. apply mul_mod_idemp_r; auto'. Qed.
+
+Theorem mul_mod: forall a b n, n~=0 ->
+ (a * b) mod n == ((a mod n) * (b mod n)) mod n.
+Proof. intros. apply mul_mod; auto'. Qed.
+
+Lemma add_mod_idemp_l : forall a b n, n~=0 ->
+ ((a mod n)+b) mod n == (a+b) mod n.
+Proof. intros. apply add_mod_idemp_l; auto'. Qed.
+
+Lemma add_mod_idemp_r : forall a b n, n~=0 ->
+ (a+(b mod n)) mod n == (a+b) mod n.
+Proof. intros. apply add_mod_idemp_r; auto'. Qed.
+
+Theorem add_mod: forall a b n, n~=0 ->
+ (a+b) mod n == (a mod n + b mod n) mod n.
+Proof. intros. apply add_mod; auto'. Qed.
+
+Lemma div_div : forall a b c, b~=0 -> c~=0 ->
+ (a/b)/c == a/(b*c).
+Proof. intros. apply div_div; auto'. Qed.
+
+(** A last inequality: *)
+
+Theorem div_mul_le:
+ forall a b c, b~=0 -> c*(a/b) <= (c*a)/b.
+Proof. intros. apply div_mul_le; auto'. Qed.
+
+(** mod is related to divisibility *)
+
+Lemma mod_divides : forall a b, b~=0 ->
+ (a mod b == 0 <-> exists c, a == b*c).
+Proof. intros. apply mod_divides; auto'. Qed.
+*)
+*)
+
+End ZDivPropFunct.
+
diff --git a/theories/Numbers/Integer/Abstract/ZDivOcaml.v b/theories/Numbers/Integer/Abstract/ZDivOcaml.v
new file mode 100644
index 0000000000..2f68da9330
--- /dev/null
+++ b/theories/Numbers/Integer/Abstract/ZDivOcaml.v
@@ -0,0 +1,541 @@
+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+
+(** Euclidean Division for integers
+
+    We use here the convention of Ocaml and many other system (C, ASM, ...),
+    i.e. Round-Toward-Zero :
+
+    [a = bq+r /\ 0 < |r| < |b| /\ Sign(r) = Sgn(a)]
+*)
+
+Require Import ZAxioms ZProperties NZDiv.
+
+Open Scope NumScope.
+
+Module Type ZDiv (Import Z : ZAxiomsSig).
+
+ Parameter Inline div : t -> t -> t.
+ Parameter Inline modulo : t -> t -> t.
+
+ Infix "/" := div : NumScope.
+ Infix "mod" := modulo (at level 40, no associativity) : NumScope.
+
+ Instance div_wd : Proper (eq==>eq==>eq) div.
+ Instance mod_wd : Proper (eq==>eq==>eq) modulo.
+
+ Axiom div_mod : forall a b, b ~= 0 -> a == b*(a/b) + (a mod b).
+ Axiom mod_bound : forall a b, 0<=a -> 0<b -> 0 <= a mod b < b.
+ Axiom mod_opp_l : forall a b, b ~= 0 -> (-a) mod b == - (a mod b).
+ Axiom mod_opp_r : forall a b, b ~= 0 -> a mod (-b) == a mod b.
+
+End ZDiv.
+
+Module Type ZDivSig := ZAxiomsSig <+ ZDiv.
+
+Module ZDivPropFunct (Import Z : ZDivSig).
+ (* TODO: en faire un arg du foncteur + comprendre le bug de SearchAbout *)
+ Module Import ZP := ZPropFunct Z.
+
+(** We benefit from what already exists for NZ *)
+
+ Module Import NZDivP := NZDivPropFunct Z.
+
+Ltac pos_or_neg a :=
+ let LT := fresh "LT" in
+ let LE := fresh "LE" in
+ destruct (le_gt_cases 0 a) as [LE|LT]; [|rewrite <- opp_pos_neg in LT].
+
+(** Another formulation of the main equation *)
+
+Lemma mod_eq :
+ forall a b, b~=0 -> a mod b == a - b*(a/b).
+Proof.
+intros.
+rewrite <- add_move_l.
+symmetry. apply div_mod; auto.
+Qed.
+
+(** A few sign rules (simple ones) *)
+
+Lemma mod_opp_opp : forall a b, b ~= 0 -> (-a) mod (-b) == - (a mod b).
+Proof. intros. rewrite mod_opp_r, mod_opp_l; auto. Qed.
+
+Lemma div_opp_l : forall a b, b ~= 0 -> (-a)/b == -(a/b).
+Proof.
+intros.
+rewrite <- (mul_cancel_l _ _ b); auto.
+rewrite <- (add_cancel_r _ _ ((-a) mod b)); auto.
+rewrite <- div_mod; auto.
+rewrite mod_opp_l, mul_opp_r, <- opp_add_distr, <- div_mod; auto.
+Qed.
+
+Lemma div_opp_r : forall a b, b ~= 0 -> a/(-b) == -(a/b).
+Proof.
+intros.
+assert (-b ~= 0) by (rewrite eq_opp_l, opp_0; auto).
+rewrite <- (mul_cancel_l _ _ (-b)); auto.
+rewrite <- (add_cancel_r _ _ (a mod (-b))); auto.
+rewrite <- div_mod; auto.
+rewrite mod_opp_r, mul_opp_opp, <- div_mod; auto.
+Qed.
+
+Lemma div_opp_opp : forall a b, b ~= 0 -> (-a)/(-b) == a/b.
+Proof. intros. rewrite div_opp_r, div_opp_l, opp_involutive; auto. Qed.
+
+(** The sign of [a mod b] is the one of [a] *)
+
+(* TODO: a proper sgn function and theory *)
+
+Lemma mod_sign : forall a b, b~=0 -> 0 <= (a mod b) * a.
+Proof.
+assert (Aux : forall a b, 0<b -> 0 <= (a mod b) * a).
+ intros. pos_or_neg a.
+ apply mul_nonneg_nonneg; auto. destruct (mod_bound a b); auto.
+ rewrite <- mul_opp_opp, <- mod_opp_l by order.
+ apply mul_nonneg_nonneg; try order. destruct (mod_bound (-a) b); try order.
+intros. pos_or_neg b. apply Aux; order.
+rewrite <- mod_opp_r by order. apply Aux; order.
+Qed.
+
+
+(** Uniqueness theorems *)
+
+Theorem div_mod_unique : forall b q1 q2 r1 r2 : t,
+  (0<=r1<b \/ b<r1<=0) -> (0<=r2<b \/ b<r2<=0) ->
+  b*q1+r1 == b*q2+r2 -> q1 == q2 /\ r1 == r2.
+Proof.
+intros b q1 q2 r1 r2 Hr1 Hr2 EQ.
+destruct Hr1; destruct Hr2; try (intuition; order).
+apply div_mod_unique with b; auto.
+rewrite <- opp_inj_wd in EQ.
+rewrite 2 opp_add_distr in EQ. rewrite <- 2 mul_opp_l in EQ.
+rewrite <- (opp_inj_wd r1 r2).
+apply div_mod_unique with (-b); auto.
+rewrite <- opp_lt_mono, opp_nonneg_nonpos; intuition.
+rewrite <- opp_lt_mono, opp_nonneg_nonpos; intuition.
+Qed.
+
+Theorem div_unique:
+ forall a b q r, 0<=a -> 0<=r<b -> a == b*q + r -> q == a/b.
+Proof. intros; apply div_unique with r; auto. Qed.
+
+Theorem mod_unique:
+ forall a b q r, 0<=a -> 0<=r<b -> a == b*q + r -> r == a mod b.
+Proof. intros; apply mod_unique with q; auto. Qed.
+
+(** A division by itself returns 1 *)
+
+Lemma div_same : forall a, a~=0 -> a/a == 1.
+Proof.
+intros. pos_or_neg a. apply div_same; order.
+rewrite <- div_opp_opp; auto. apply div_same; auto.
+Qed.
+
+Lemma mod_same : forall a, a~=0 -> a mod a == 0.
+Proof.
+intros. rewrite mod_eq, div_same; auto. nzsimpl. apply sub_diag.
+Qed.
+
+(** A division of a small number by a bigger one yields zero. *)
+
+Theorem div_small: forall a b, 0<=a<b -> a/b == 0.
+Proof. exact div_small. Qed.
+
+(** Same situation, in term of modulo: *)
+
+Theorem mod_small: forall a b, 0<=a<b -> a mod b == a.
+Proof. exact mod_small. Qed.
+
+(** * Basic values of divisions and modulo. *)
+
+Lemma div_0_l: forall a, a~=0 -> 0/a == 0.
+Proof.
+intros. pos_or_neg a. apply div_0_l; order.
+rewrite <- div_opp_opp, opp_0; auto. apply div_0_l; auto.
+Qed.
+
+Lemma mod_0_l: forall a, a~=0 -> 0 mod a == 0.
+Proof.
+intros; rewrite mod_eq, div_0_l; nzsimpl; auto.
+Qed.
+
+Lemma div_1_r: forall a, a/1 == a.
+Proof.
+intros. pos_or_neg a. apply div_1_r; auto.
+apply opp_inj. rewrite <- div_opp_l. apply div_1_r; order.
+intro EQ; symmetry in EQ; revert EQ; apply lt_neq, lt_0_1.
+Qed.
+
+Lemma mod_1_r: forall a, a mod 1 == 0.
+Proof.
+intros. rewrite mod_eq, div_1_r; nzsimpl; auto using sub_diag.
+intro EQ; symmetry in EQ; revert EQ; apply lt_neq; apply lt_0_1.
+Qed.
+
+Lemma div_1_l: forall a, 1<a -> 1/a == 0.
+Proof. exact div_1_l. Qed.
+
+Lemma mod_1_l: forall a, 1<a -> 1 mod a == 1.
+Proof. exact mod_1_l. Qed.
+
+Lemma div_mul : forall a b, b~=0 -> (a*b)/b == a.
+Proof.
+intros. pos_or_neg a; pos_or_neg b. apply div_mul; order.
+rewrite <- div_opp_opp, <- mul_opp_r by order. apply div_mul; order.
+rewrite <- opp_inj_wd, <- div_opp_l, <- mul_opp_l by order. apply div_mul; order.
+rewrite <- opp_inj_wd, <- div_opp_r, <- mul_opp_opp by order. apply div_mul; order.
+Qed.
+
+Lemma mod_mul : forall a b, b~=0 -> (a*b) mod b == 0.
+Proof.
+intros. rewrite mod_eq, div_mul; auto. rewrite mul_comm; apply sub_diag.
+Qed.
+
+(** * Order results about mod and div *)
+
+(** A modulo cannot grow beyond its starting point. *)
+
+Theorem mod_le: forall a b, 0<=a -> 0<b -> a mod b <= a.
+Proof. exact mod_le. Qed.
+
+Theorem div_pos : forall a b, 0<=a -> 0<b -> 0<= a/b.
+Proof. exact div_pos. Qed.
+
+Lemma div_str_pos : forall a b, 0<b<=a -> 0 < a/b.
+Proof. exact div_str_pos. Qed.
+
+(** TODO: TO MIGRATE LATER *)
+Definition abs z := max z (-z).
+Lemma abs_pos : forall z, 0<=z -> abs z == z.
+Proof.
+intros; apply max_l. apply le_trans with 0; auto.
+rewrite opp_nonpos_nonneg; auto.
+Qed.
+Lemma abs_neg : forall z, 0<=-z -> abs z == -z.
+Proof.
+intros; apply max_r. apply le_trans with 0; auto.
+rewrite <- opp_nonneg_nonpos; auto.
+Qed.
+
+Lemma eq_sym_iff : forall x y, x==y <-> y==x.
+Proof.
+intros; split; symmetry; auto.
+Qed.
+
+(** END TODO *)
+
+Lemma div_small_iff : forall a b, b~=0 -> (a/b==0 <-> abs a < abs b).
+Proof.
+intros. pos_or_neg a; pos_or_neg b.
+rewrite div_small_iff; try order. rewrite 2 abs_pos; intuition; order.
+rewrite <- opp_inj_wd, opp_0, <- div_opp_r, div_small_iff by order.
+ rewrite (abs_pos a), (abs_neg b); intuition; order.
+rewrite <- opp_inj_wd, opp_0, <- div_opp_l, div_small_iff by order.
+ rewrite (abs_neg a), (abs_pos b); intuition; order.
+rewrite <- div_opp_opp, div_small_iff by order.
+ rewrite (abs_neg a), (abs_neg b); intuition; order.
+Qed.
+
+Lemma mod_small_iff : forall a b, b~=0 -> (a mod b == a <-> abs a < abs b).
+Proof.
+intros. rewrite mod_eq, <- div_small_iff by order.
+rewrite sub_move_r, <- (add_0_r a) at 1. rewrite add_cancel_l.
+rewrite eq_sym_iff, eq_mul_0. intuition.
+Qed.
+
+(** As soon as the divisor is strictly greater than 1,
+    the division is strictly decreasing. *)
+
+Lemma div_lt : forall a b, 0<a -> 1<b -> a/b < a.
+Proof. exact div_lt. Qed.
+
+(** [le] is compatible with a positive division. *)
+
+Lemma div_le_mono : forall a b c, 0<c -> a<=b -> a/c <= b/c.
+Proof.
+intros. pos_or_neg a. apply div_le_mono; auto.
+pos_or_neg b. apply le_trans with 0.
+ rewrite <- opp_nonneg_nonpos, <- div_opp_l by order.
+ apply div_pos; order.
+ apply div_pos; order.
+rewrite opp_le_mono in *. rewrite <- 2 div_opp_l by order.
+ apply div_le_mono; intuition; order.
+Qed.
+
+(** With this choice of division,
+    rounding of div is always done toward zero: *)
+
+Lemma mul_div_le : forall a b, 0<=a -> b~=0 -> 0 <= b*(a/b) <= a.
+Proof.
+intros. pos_or_neg b.
+split.
+apply mul_nonneg_nonneg; [|apply div_pos]; order.
+apply mul_div_le; order.
+rewrite <- mul_opp_opp, <- div_opp_r by order.
+split.
+apply mul_nonneg_nonneg; [|apply div_pos]; order.
+apply mul_div_le; order.
+Qed.
+
+Lemma mul_div_ge : forall a b, a<=0 -> b~=0 -> a <= b*(a/b) <= 0.
+Proof.
+intros.
+rewrite <- opp_nonneg_nonpos, opp_le_mono, <-mul_opp_r, <-div_opp_l by order.
+destruct (mul_div_le (-a) b); auto.
+rewrite opp_nonneg_nonpos; auto.
+Qed.
+
+(** For positive numbers, considering [S (a/b)] leads to an upper bound for [a] *)
+
+Lemma mul_succ_div_gt: forall a b, 0<=a -> 0<b -> a < b*(S (a/b)).
+Proof. exact mul_succ_div_gt. Qed.
+
+(** Some previous inequalities are exact iff the modulo is zero. *)
+
+Lemma div_exact : forall a b, b~=0 -> (a == b*(a/b) <-> a mod b == 0).
+Proof.
+intros. rewrite mod_eq by order. rewrite sub_move_r; nzsimpl; intuition.
+Qed.
+
+(** Some additionnal inequalities about div. *)
+
+Theorem div_lt_upper_bound:
+  forall a b q, 0<=a -> 0<b -> a < b*q -> a/b < q.
+Proof. exact div_lt_upper_bound. Qed.
+
+Theorem div_le_upper_bound:
+  forall a b q, 0<b -> a <= b*q -> a/b <= q.
+Proof.
+intros.
+rewrite <- (div_mul q b) by order.
+apply div_le_mono; auto. rewrite mul_comm; auto.
+Qed.
+
+Theorem div_le_lower_bound:
+  forall a b q, 0<b -> b*q <= a -> q <= a/b.
+Proof.
+intros.
+rewrite <- (div_mul q b) by order.
+apply div_le_mono; auto. rewrite mul_comm; auto.
+Qed.
+
+(** A division respects opposite monotonicity for the divisor *)
+
+Lemma div_le_compat_l: forall p q r, 0<=p -> 0<q<=r -> p/r <= p/q.
+Proof. exact div_le_compat_l. Qed.
+
+(** * Relations between usual operations and mod and div *)
+
+(** Unlike with other division conventions, some results here aren't
+    always valid, and need to be restricted. For instance
+    [(a+b*c) mod c <> a mod c] for [a=9,b=-5,c=2] *)
+
+Lemma mod_add : forall a b c, c~=0 -> 0 <= (a+b*c)*a ->
+ (a + b * c) mod c == a mod c.
+Proof.
+assert (forall a b c, c~=0 -> 0<=a -> 0<=a+b*c -> (a+b*c) mod c == a mod c).
+ intros. pos_or_neg c. apply mod_add; order.
+ rewrite <- (mod_opp_r a), <- (mod_opp_r (a+b*c)) by order.
+ rewrite <- mul_opp_opp in *.
+ apply mod_add; order.
+intros a b c Hc Habc.
+destruct (le_0_mul _ _ Habc) as [(Habc',Ha)|(Habc',Ha)]; auto.
+apply opp_inj. revert Ha Habc'.
+rewrite <- 2 opp_nonneg_nonpos.
+rewrite <- 2 mod_opp_l, opp_add_distr, <- mul_opp_l by order; auto.
+Qed.
+
+Lemma div_add : forall a b c, c~=0 -> 0 <= (a+b*c)*a ->
+ (a + b * c) / c == a / c + b.
+Proof.
+intros.
+rewrite <- (mul_cancel_l _ _ c) by auto.
+rewrite <- (add_cancel_r _ _ ((a+b*c) mod c)).
+rewrite <- div_mod; auto.
+rewrite mod_add; auto.
+rewrite mul_add_distr_l, add_shuffle0, <-div_mod, mul_comm; auto.
+Qed.
+
+Lemma div_add_l: forall a b c, b~=0 -> 0 <= (a*b+c)*c ->
+ (a * b + c) / b == a + c / b.
+Proof.
+ intros a b c. rewrite add_comm, (add_comm a). intros; apply div_add; auto.
+Qed.
+
+(** Cancellations. *)
+
+Lemma div_mul_cancel_r : forall a b c, b~=0 -> c~=0 ->
+ (a*c)/(b*c) == a/b.
+Proof.
+assert (Aux1 : forall a b c, 0<=a -> 0<b -> c~=0 -> (a*c)/(b*c) == a/b).
+ intros. pos_or_neg c. apply div_mul_cancel_r; order.
+ rewrite <- div_opp_opp, <- 2 mul_opp_r. apply div_mul_cancel_r; order.
+ rewrite <- neq_mul_0; intuition order.
+assert (Aux2 : forall a b c, 0<=a -> b~=0 -> c~=0 -> (a*c)/(b*c) == a/b).
+ intros. pos_or_neg b. apply Aux1; order.
+ apply opp_inj. rewrite <- 2 div_opp_r, <- mul_opp_l; try order. apply Aux1; order.
+ rewrite <- neq_mul_0; intuition order.
+intros. pos_or_neg a. apply Aux2; order.
+apply opp_inj. rewrite <- 2 div_opp_l, <- mul_opp_l; try order. apply Aux2; order.
+rewrite <- neq_mul_0; intuition order.
+Qed.
+
+Lemma div_mul_cancel_l : forall a b c, b~=0 -> c~=0 ->
+ (c*a)/(c*b) == a/b.
+Proof.
+intros. rewrite !(mul_comm c); apply div_mul_cancel_r; auto.
+Qed.
+
+Lemma mul_mod_distr_r: forall a b c, b~=0 -> c~=0 ->
+  (a*c) mod (b*c) == (a mod b) * c.
+Proof.
+intros.
+assert (b*c ~= 0) by (rewrite <- neq_mul_0; intuition).
+rewrite ! mod_eq by auto.
+rewrite div_mul_cancel_r by order.
+rewrite mul_sub_distr_r, <- !mul_assoc, (mul_comm (a/b) c); auto.
+Qed.
+
+Lemma mul_mod_distr_l: forall a b c, b~=0 -> c~=0 ->
+  (c*a) mod (c*b) == c * (a mod b).
+Proof.
+intros; rewrite !(mul_comm c); apply mul_mod_distr_r; auto.
+Qed.
+
+(** Operations modulo. *)
+
+Theorem mod_mod: forall a n, n~=0 ->
+ (a mod n) mod n == a mod n.
+Proof.
+intros. pos_or_neg a; pos_or_neg n. apply mod_mod; order.
+rewrite <- ! (mod_opp_r _ n) by auto. apply mod_mod; order.
+apply opp_inj. rewrite <- !mod_opp_l by order. apply mod_mod; order.
+apply opp_inj. rewrite <- !mod_opp_opp by order. apply mod_mod; order.
+Qed.
+
+Lemma mul_mod_idemp_l : forall a b n, n~=0 ->
+ ((a mod n)*b) mod n == (a*b) mod n.
+Proof.
+assert (Aux1 : forall a b n, 0<=a -> 0<=b -> n~=0 ->
+         ((a mod n)*b) mod n == (a*b) mod n).
+ intros. pos_or_neg n. apply mul_mod_idemp_l; order.
+ rewrite <- ! (mod_opp_r _ n) by order. apply mul_mod_idemp_l; order.
+assert (Aux2 : forall a b n, 0<=a -> n~=0 ->
+         ((a mod n)*b) mod n == (a*b) mod n).
+ intros. pos_or_neg b. apply Aux1; auto.
+ apply opp_inj. rewrite <-2 mod_opp_l, <-2 mul_opp_r by order.
+  apply Aux1; order.
+intros a b n Hn. pos_or_neg a. apply Aux2; auto.
+apply opp_inj. rewrite <-2 mod_opp_l, <-2 mul_opp_l, <-mod_opp_l by order.
+apply Aux2; order.
+Qed.
+
+Lemma mul_mod_idemp_r : forall a b n, n~=0 ->
+ (a*(b mod n)) mod n == (a*b) mod n.
+Proof.
+intros. rewrite !(mul_comm a). apply mul_mod_idemp_l; auto.
+Qed.
+
+Theorem mul_mod: forall a b n, n~=0 ->
+ (a * b) mod n == ((a mod n) * (b mod n)) mod n.
+Proof.
+intros. rewrite mul_mod_idemp_l, mul_mod_idemp_r; auto.
+Qed.
+
+(** addition and modulo
+
+  Generally speaking, unlike with other conventions, we don't have
+       [(a+b) mod n = (a mod n + b mod n) mod n]
+  for any a and b.
+  For instance, take (8 + (-10)) mod 3 = -2 whereas
+  (8 mod 3 + (-10 mod 3)) mod 3 = 1.
+*)
+
+Lemma add_mod_idemp_l : forall a b n, n~=0 -> 0 <= a*b ->
+ ((a mod n)+b) mod n == (a+b) mod n.
+Proof.
+assert (Aux : forall a b n, 0<=a -> 0<=b -> n~=0 ->
+          ((a mod n)+b) mod n == (a+b) mod n).
+ intros. pos_or_neg n. apply add_mod_idemp_l; order.
+ rewrite <- ! (mod_opp_r _ n) by order. apply add_mod_idemp_l; order.
+intros a b n Hn Hab. destruct (le_0_mul _ _ Hab) as [(Ha,Hb)|(Ha,Hb)].
+apply Aux; auto.
+apply opp_inj. rewrite <-2 mod_opp_l, 2 opp_add_distr, <-mod_opp_l by order.
+apply Aux; auto.
+rewrite opp_nonneg_nonpos; auto.
+rewrite opp_nonneg_nonpos; auto.
+Qed.
+
+Lemma add_mod_idemp_r : forall a b n, n~=0 -> 0 <= a*b ->
+ (a+(b mod n)) mod n == (a+b) mod n.
+Proof.
+intros. rewrite !(add_comm a). apply add_mod_idemp_l; auto.
+rewrite mul_comm; auto.
+Qed.
+
+Theorem add_mod: forall a b n, n~=0 -> 0 <= a*b ->
+ (a+b) mod n == (a mod n + b mod n) mod n.
+Proof.
+intros a b n Hn Hab. rewrite add_mod_idemp_l, add_mod_idemp_r; auto.
+destruct (le_0_mul _ _ Hab) as [(Ha,Hb)|(Ha,Hb)];
+ destruct (le_0_mul _ _ (mod_sign b n Hn)) as [(Hb',Hm)|(Hb',Hm)];
+ auto using mul_nonneg_nonneg, mul_nonpos_nonpos.
+ setoid_replace b with 0 by order. rewrite mod_0_l by order. nzsimpl; order.
+ setoid_replace b with 0 by order. rewrite mod_0_l by order. nzsimpl; order.
+Qed.
+
+
+(** Conversely, the following result needs less restrictions here. *)
+
+Lemma div_div : forall a b c, b~=0 -> c~=0 ->
+ (a/b)/c == a/(b*c).
+Proof.
+assert (Aux1 : forall a b c, 0<=a -> 0<b -> c~=0 -> (a/b)/c == a/(b*c)).
+ intros. pos_or_neg c. apply div_div; order.
+ apply opp_inj. rewrite <- 2 div_opp_r, <- mul_opp_r; auto. apply div_div; order.
+ rewrite <- neq_mul_0; intuition order.
+assert (Aux2 : forall a b c, 0<=a -> b~=0 -> c~=0 -> (a/b)/c == a/(b*c)).
+ intros. pos_or_neg b. apply Aux1; order.
+ apply opp_inj. rewrite <- div_opp_l, <- 2 div_opp_r, <- mul_opp_l; auto.
+ rewrite <- neq_mul_0; intuition order.
+intros. pos_or_neg a. apply Aux2; order.
+apply opp_inj. rewrite <- 3 div_opp_l; try order. apply Aux2; order.
+rewrite <- neq_mul_0; intuition order.
+Qed.
+
+(** A last inequality: *)
+
+Theorem div_mul_le:
+ forall a b c, 0<=a -> 0<b -> 0<=c -> c*(a/b) <= (c*a)/b.
+Proof. exact div_mul_le. Qed.
+
+(** mod is related to divisibility *)
+
+Lemma mod_divides : forall a b, b~=0 ->
+ (a mod b == 0 <-> exists c, a == b*c).
+Proof.
+ intros.
+ pos_or_neg a; pos_or_neg b.
+ apply mod_divides; order.
+ rewrite <- mod_opp_r, mod_divides by order.
+  split; intros (c,Hc); exists (-c).
+  rewrite mul_opp_r, <- mul_opp_l; auto.
+  rewrite mul_opp_opp; auto.
+ rewrite <- opp_inj_wd, opp_0, <- mod_opp_l, mod_divides by order.
+  split; intros (c,Hc); exists (-c).
+  rewrite mul_opp_r, eq_opp_r; auto.
+  rewrite mul_opp_r, opp_inj_wd; auto.
+ rewrite <- opp_inj_wd, opp_0, <- mod_opp_opp, mod_divides by order.
+  split; intros (c,Hc); exists c.
+  rewrite <-opp_inj_wd, <- mul_opp_l; auto.
+  rewrite mul_opp_l, opp_inj_wd; auto.
+Qed.
+
+End ZDivPropFunct.
+
diff --git a/theories/Numbers/Integer/Abstract/ZMulOrder.v b/theories/Numbers/Integer/Abstract/ZMulOrder.v
index 4f11dcc5c0..a9cf0dc4d3 100644
--- a/theories/Numbers/Integer/Abstract/ZMulOrder.v
+++ b/theories/Numbers/Integer/Abstract/ZMulOrder.v
@@ -34,12 +34,6 @@ now apply mul_le_mono_nonpos_l.
 apply mul_le_mono_nonpos_r; [now apply le_trans with q | assumption].
 Qed.
 
-Theorem mul_nonneg_nonneg : forall n m, 0 <= n -> 0 <= m -> 0 <= n * m.
-Proof.
-intros n m H1 H2.
-rewrite <- (mul_0_l m). now apply mul_le_mono_nonneg_r.
-Qed.
-
 Theorem mul_nonpos_nonpos : forall n m, n <= 0 -> m <= 0 -> 0 <= n * m.
 Proof.
 intros n m H1 H2.
diff --git a/theories/Numbers/NatInt/NZAdd.v b/theories/Numbers/NatInt/NZAdd.v
index 579d2197f5..785abb5c7f 100644
--- a/theories/Numbers/NatInt/NZAdd.v
+++ b/theories/Numbers/NatInt/NZAdd.v
@@ -67,18 +67,20 @@ Proof.
 intros n m p. rewrite (add_comm n p), (add_comm m p). apply add_cancel_l.
 Qed.
 
+Theorem add_shuffle0 : forall n m p, n+m+p == n+p+m.
+Proof.
+intros n m p. rewrite <- 2 add_assoc, add_cancel_l. apply add_comm.
+Qed.
+
 Theorem add_shuffle1 : forall n m p q, (n + m) + (p + q) == (n + p) + (m + q).
 Proof.
-intros n m p q.
-rewrite <- (add_assoc n m), <- (add_assoc n p), add_cancel_l.
-rewrite 2 add_assoc, add_cancel_r. now apply add_comm.
+intros n m p q. rewrite 2 add_assoc, add_cancel_r. apply add_shuffle0.
 Qed.
 
 Theorem add_shuffle2 : forall n m p q, (n + m) + (p + q) == (n + q) + (m + p).
 Proof.
 intros n m p q.
-rewrite <- (add_assoc n m), <- (add_assoc n q), add_cancel_l.
-rewrite add_assoc. now apply add_comm.
+rewrite 2 add_assoc, add_shuffle0, add_cancel_r. apply add_shuffle0.
 Qed.
 
 Theorem sub_1_r : forall n, n - 1 == P n.
diff --git a/theories/Numbers/NatInt/NZAxioms.v b/theories/Numbers/NatInt/NZAxioms.v
index 20a85e2ef5..c0f3093291 100644
--- a/theories/Numbers/NatInt/NZAxioms.v
+++ b/theories/Numbers/NatInt/NZAxioms.v
@@ -102,6 +102,11 @@ Notation "x <= y" := (le x y) : NumScope.
 Notation "x > y" := (lt y x) (only parsing) : NumScope.
 Notation "x >= y" := (le y x) (only parsing) : NumScope.
 
+Notation "x < y < z" := (x<y /\ y<z) : NumScope.
+Notation "x <= y <= z" := (x<=y /\ y<=z) : NumScope.
+Notation "x <= y < z" := (x<=y /\ y<z) : NumScope.
+Notation "x < y <= z" := (x<y /\ y<=z) : NumScope.
+
 Instance lt_wd : Proper (eq ==> eq ==> iff) lt.
 (** Compatibility of [le] can be proved later from [lt_wd]
     and [lt_eq_cases] *)
diff --git a/theories/Numbers/NatInt/NZDiv.v b/theories/Numbers/NatInt/NZDiv.v
new file mode 100644
index 0000000000..9ea654cc92
--- /dev/null
+++ b/theories/Numbers/NatInt/NZDiv.v
@@ -0,0 +1,528 @@
+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+
+(** Euclidean Division *)
+
+Require Import NZAxioms NZMulOrder.
+
+Open Scope NumScope.
+
+Module Type NZDiv (Import NZ : NZOrdAxiomsSig).
+
+ Parameter Inline div : t -> t -> t.
+ Parameter Inline modulo : t -> t -> t.
+
+ Infix "/" := div : NumScope.
+ Infix "mod" := modulo (at level 40, no associativity) : NumScope.
+
+ Instance div_wd : Proper (eq==>eq==>eq) div.
+ Instance mod_wd : Proper (eq==>eq==>eq) modulo.
+
+ Axiom div_mod : forall a b, b ~= 0 -> a == b*(a/b) + (a mod b).
+ Axiom mod_bound : forall a b, 0<=a -> 0<b -> 0 <= a mod b < b.
+
+End NZDiv.
+
+Module Type NZDivSig := NZOrdAxiomsSig <+ NZDiv.
+
+Module NZDivPropFunct (Import NZ : NZDivSig).
+ (* TODO: a transformer un jour en arg funct puis include *)
+ Module Import P := NZMulOrderPropFunct NZ.
+
+(** Uniqueness theorems *)
+
+Theorem div_mod_unique :
+ forall b q1 q2 r1 r2, 0<=r1<b -> 0<=r2<b ->
+  b*q1+r1 == b*q2+r2 -> q1 == q2 /\ r1 == r2.
+Proof.
+intros b.
+assert (U : forall q1 q2 r1 r2,
+            b*q1+r1 == b*q2+r2 -> 0<=r1<b -> 0<=r2 -> q1<q2 -> False).
+ intros q1 q2 r1 r2 EQ LT Hr1 Hr2.
+ contradict EQ.
+ apply lt_neq.
+ apply lt_le_trans with (b*q1+b).
+ rewrite <- add_lt_mono_l; intuition.
+ apply le_trans with (b*q2).
+ rewrite mul_comm, <- mul_succ_l, mul_comm.
+ apply mul_le_mono_nonneg_l; intuition; try order.
+ rewrite le_succ_l; auto.
+ rewrite <- (add_0_r (b*q2)) at 1.
+ rewrite <- add_le_mono_l; intuition.
+
+intros q1 q2 r1 r2 Hr1 Hr2 EQ; destruct (lt_trichotomy q1 q2) as [LT|[EQ'|GT]].
+elim (U q1 q2 r1 r2); intuition.
+split; auto. rewrite EQ' in EQ. rewrite add_cancel_l in EQ; auto.
+elim (U q2 q1 r2 r1); intuition.
+Qed.
+
+Theorem div_unique:
+ forall a b q r, 0<=a -> 0<=r<b ->
+   a == b*q + r -> q == a/b.
+Proof.
+intros a b q r Ha (Hb,Hr) EQ.
+rewrite (div_mod a b) in EQ by order.
+destruct (div_mod_unique b (a/b) q (a mod b) r); auto.
+apply mod_bound; order.
+Qed.
+
+Theorem mod_unique:
+ forall a b q r, 0<=a -> 0<=r<b ->
+  a == b*q + r -> r == a mod b.
+Proof.
+intros a b q r Ha (Hb,Hr) EQ.
+rewrite (div_mod a b) in EQ by order.
+destruct (div_mod_unique b (a/b) q (a mod b) r); auto.
+apply mod_bound; order.
+Qed.
+
+
+(** A division by itself returns 1 *)
+
+Lemma div_same : forall a, 0<a -> a/a == 1.
+Proof.
+intros. symmetry.
+apply div_unique with 0; intuition; try order.
+nzsimpl; auto.
+Qed.
+
+Lemma mod_same : forall a, 0<a -> a mod a == 0.
+Proof.
+intros. symmetry.
+apply mod_unique with 1; intuition; try order.
+nzsimpl; auto.
+Qed.
+
+(** A division of a small number by a bigger one yields zero. *)
+
+Theorem div_small: forall a b, 0<=a<b -> a/b == 0.
+Proof.
+intros. symmetry.
+apply div_unique with a; intuition; try order.
+nzsimpl; auto.
+Qed.
+
+(** Same situation, in term of modulo: *)
+
+Theorem mod_small: forall a b, 0<=a<b -> a mod b == a.
+Proof.
+intros. symmetry.
+apply mod_unique with 0; intuition; try order.
+nzsimpl; auto.
+Qed.
+
+(** * Basic values of divisions and modulo. *)
+
+Lemma div_0_l: forall a, 0<a -> 0/a == 0.
+Proof.
+intros; apply div_small; split; order.
+Qed.
+
+Lemma mod_0_l: forall a, 0<a -> 0 mod a == 0.
+Proof.
+intros; apply mod_small; split; order.
+Qed.
+
+Lemma div_1_r: forall a, 0<=a -> a/1 == a.
+Proof.
+intros. symmetry.
+apply div_unique with 0; try split; try order; try apply lt_0_1.
+nzsimpl; auto.
+Qed.
+
+Lemma mod_1_r: forall a, 0<=a -> a mod 1 == 0.
+Proof.
+intros. symmetry.
+apply mod_unique with a; try split; try order; try apply lt_0_1.
+nzsimpl; auto.
+Qed.
+
+Lemma div_1_l: forall a, 1<a -> 1/a == 0.
+Proof.
+intros; apply div_small; split; auto. apply le_succ_diag_r.
+Qed.
+
+Lemma mod_1_l: forall a, 1<a -> 1 mod a == 1.
+Proof.
+intros; apply mod_small; split; auto. apply le_succ_diag_r.
+Qed.
+
+Lemma div_mul : forall a b, 0<=a -> 0<b -> (a*b)/b == a.
+Proof.
+intros; symmetry.
+apply div_unique with 0; try split; try order.
+apply mul_nonneg_nonneg; order.
+nzsimpl; apply mul_comm.
+Qed.
+
+Lemma mod_mul : forall a b, 0<=a -> 0<b -> (a*b) mod b == 0.
+Proof.
+intros; symmetry.
+apply mod_unique with a; try split; try order.
+apply mul_nonneg_nonneg; order.
+nzsimpl; apply mul_comm.
+Qed.
+
+
+(** * Order results about mod and div *)
+
+(** A modulo cannot grow beyond its starting point. *)
+
+Theorem mod_le: forall a b, 0<=a -> 0<b -> a mod b <= a.
+Proof.
+intros. destruct (le_gt_cases b a).
+apply le_trans with b; auto.
+apply lt_le_incl. destruct (mod_bound a b); auto.
+rewrite lt_eq_cases; right.
+apply mod_small; auto.
+Qed.
+
+
+(* Division of positive numbers is positive. *)
+
+Lemma div_pos: forall a b, 0<=a -> 0<b -> 0 <= a/b.
+Proof.
+intros.
+rewrite (mul_le_mono_pos_l _ _ b); auto; nzsimpl.
+rewrite (add_le_mono_r _ _ (a mod b)).
+rewrite <- div_mod by order.
+nzsimpl.
+apply mod_le; auto.
+Qed.
+
+Lemma div_str_pos : forall a b, 0<b<=a -> 0 < a/b.
+Proof.
+intros a b (Hb,Hab).
+assert (LE : 0 <= a/b) by (apply div_pos; order).
+assert (MOD : a mod b < b) by (destruct (mod_bound a b); order).
+rewrite lt_eq_cases in LE; destruct LE as [LT|EQ]; auto.
+exfalso; revert Hab.
+rewrite (div_mod a b), <-EQ; nzsimpl; order.
+Qed.
+
+Lemma div_small_iff : forall a b, 0<=a -> 0<b -> (a/b==0 <-> a<b).
+Proof.
+intros a b Ha Hb; split; intros Hab.
+destruct (lt_ge_cases a b); auto.
+symmetry in Hab. contradict Hab. apply lt_neq, div_str_pos; auto.
+apply div_small; auto.
+Qed.
+
+Lemma mod_small_iff : forall a b, 0<=a -> 0<b -> (a mod b == a <-> a<b).
+Proof.
+intros a b Ha Hb. split; intros H; auto using mod_small.
+rewrite <- div_small_iff; auto.
+rewrite <- (mul_cancel_l _ _ b) by order.
+rewrite <- (add_cancel_r _ _ (a mod b)).
+rewrite <- div_mod, H by order. nzsimpl; auto.
+Qed.
+
+Lemma div_str_pos_iff : forall a b, 0<=a -> 0<b -> (0<a/b <-> b<=a).
+Proof.
+intros a b Ha Hb; split; intros Hab.
+destruct (lt_ge_cases a b) as [LT|LE]; auto.
+rewrite <- div_small_iff in LT; order.
+apply div_str_pos; auto.
+Qed.
+
+
+(** As soon as the divisor is strictly greater than 1,
+    the division is strictly decreasing. *)
+
+Lemma div_lt : forall a b, 0<a -> 1<b -> a/b < a.
+Proof.
+intros.
+assert (0 < b) by (apply lt_trans with 1; auto using lt_0_1).
+destruct (lt_ge_cases a b).
+rewrite div_small; try split; order.
+rewrite (div_mod a b) at 2; [|order].
+apply lt_le_trans with (b*(a/b)).
+rewrite <- (mul_1_l (a/b)) at 1.
+rewrite <- mul_lt_mono_pos_r; auto.
+apply div_str_pos; auto.
+rewrite <- (add_0_r (b*(a/b))) at 1.
+rewrite <- add_le_mono_l. destruct (mod_bound a b); order.
+Qed.
+
+(** [le] is compatible with a positive division. *)
+
+Lemma div_le_mono : forall a b c, 0<c -> 0<=a<=b -> a/c <= b/c.
+Proof.
+intros a b c Hc (Ha,Hab).
+rewrite lt_eq_cases in Hab. destruct Hab as [LT|EQ];
+ [|rewrite EQ; order].
+rewrite <- lt_succ_r.
+rewrite (mul_lt_mono_pos_l c) by order.
+nzsimpl.
+rewrite (add_lt_mono_r _ _ (a mod c)).
+rewrite <- div_mod by order.
+apply lt_le_trans with b; auto.
+rewrite (div_mod b c) at 1; [| order].
+rewrite <- add_assoc, <- add_le_mono_l.
+apply le_trans with (c+0).
+nzsimpl; destruct (mod_bound b c); order.
+rewrite <- add_le_mono_l. destruct (mod_bound a c); order.
+Qed.
+
+(** The following two properties could be used as specification of div *)
+
+Lemma mul_div_le : forall a b, 0<=a -> 0<b -> b*(a/b) <= a.
+Proof.
+intros.
+rewrite (add_le_mono_r _ _ (a mod b)), <- div_mod by order.
+rewrite <- (add_0_r a) at 1.
+rewrite <- add_le_mono_l. destruct (mod_bound a b); order.
+Qed.
+
+Lemma mul_succ_div_gt : forall a b, 0<=a -> 0<b -> a < b*(S (a/b)).
+Proof.
+intros.
+rewrite (div_mod a b) at 1; [|order].
+rewrite (mul_succ_r).
+rewrite <- add_lt_mono_l.
+destruct (mod_bound a b); auto.
+Qed.
+
+
+(** The previous inequality is exact iff the modulo is zero. *)
+
+Lemma div_exact : forall a b, 0<=a -> 0<b -> (a == b*(a/b) <-> a mod b == 0).
+Proof.
+intros. rewrite (div_mod a b) at 1; [|order].
+rewrite <- (add_0_r (b*(a/b))) at 2.
+apply add_cancel_l.
+Qed.
+
+(** Some additionnal inequalities about div. *)
+
+Theorem div_lt_upper_bound:
+  forall a b q, 0<=a -> 0<b -> a < b*q -> a/b < q.
+Proof.
+intros.
+rewrite (mul_lt_mono_pos_l b) by order.
+apply le_lt_trans with a; auto.
+apply mul_div_le; auto.
+Qed.
+
+Theorem div_le_upper_bound:
+  forall a b q, 0<=a -> 0<b -> a <= b*q -> a/b <= q.
+Proof.
+intros.
+rewrite (mul_le_mono_pos_l _ _ b) by order.
+apply le_trans with a; auto.
+apply mul_div_le; auto.
+Qed.
+
+Theorem div_le_lower_bound:
+  forall a b q, 0<=a -> 0<b -> b*q <= a -> q <= a/b.
+Proof.
+intros a b q Ha Hb H.
+destruct (lt_ge_cases 0 q).
+rewrite <- (div_mul q b); try order.
+apply div_le_mono; auto.
+rewrite mul_comm; split; auto.
+apply lt_le_incl, mul_pos_pos; auto.
+apply le_trans with 0; auto; apply div_pos; auto.
+Qed.
+
+(** A division respects opposite monotonicity for the divisor *)
+
+Lemma div_le_compat_l: forall p q r, 0<=p -> 0<q<=r ->
+    p/r <= p/q.
+Proof.
+ intros p q r Hp (Hq,Hqr).
+ apply div_le_lower_bound; auto.
+ rewrite (div_mod p r) at 2; [|order].
+ apply le_trans with (r*(p/r)).
+ apply mul_le_mono_nonneg_r; try order.
+ apply div_pos; order.
+ rewrite <- (add_0_r (r*(p/r))) at 1.
+ rewrite <- add_le_mono_l. destruct (mod_bound p r); order.
+Qed.
+
+
+(** * Relations between usual operations and mod and div *)
+
+Lemma mod_add : forall a b c, 0<=a -> 0<=a+b*c -> 0<c ->
+ (a + b * c) mod c == a mod c.
+Proof.
+ intros.
+ symmetry.
+ apply mod_unique with (a/c+b); auto.
+ apply mod_bound; auto.
+ rewrite mul_add_distr_l, add_shuffle0, <- div_mod by order.
+ rewrite mul_comm; auto.
+Qed.
+
+Lemma div_add : forall a b c, 0<=a -> 0<=a+b*c -> 0<c ->
+ (a + b * c) / c == a / c + b.
+Proof.
+ intros.
+ apply (mul_cancel_l _ _ c); try order.
+ apply (add_cancel_r _ _ ((a+b*c) mod c)).
+ rewrite <- div_mod, mod_add by order.
+ rewrite mul_add_distr_l, add_shuffle0, <- div_mod by order.
+ rewrite mul_comm; auto.
+Qed.
+
+Lemma div_add_l: forall a b c, 0<=c -> 0<=a*b+c -> 0<b ->
+ (a * b + c) / b == a + c / b.
+Proof.
+ intros a b c. rewrite (add_comm _ c), (add_comm a). intros. apply div_add; auto.
+Qed.
+
+(** Cancellations. *)
+
+Lemma div_mul_cancel_r : forall a b c, 0<=a -> 0<b -> 0<c ->
+ (a*c)/(b*c) == a/b.
+Proof.
+ intros.
+ symmetry.
+ apply div_unique with ((a mod b)*c).
+ apply mul_nonneg_nonneg; order.
+ split.
+ apply mul_nonneg_nonneg; destruct (mod_bound a b); order.
+ rewrite <- mul_lt_mono_pos_r; auto. destruct (mod_bound a b); auto.
+ rewrite (div_mod a b) at 1; [|order].
+ rewrite mul_add_distr_r.
+ rewrite add_cancel_r.
+ rewrite <- 2 mul_assoc. rewrite (mul_comm c); auto.
+Qed.
+
+Lemma div_mul_cancel_l : forall a b c, 0<=a -> 0<b -> 0<c ->
+ (c*a)/(c*b) == a/b.
+Proof.
+ intros.
+ rewrite (mul_comm c a), (mul_comm c b); apply div_mul_cancel_r; auto.
+Qed.
+
+Lemma mul_mod_distr_l: forall a b c, 0<=a -> 0<b -> 0<c ->
+  (c*a) mod (c*b) == c * (a mod b).
+Proof.
+ intros.
+ rewrite <- (add_cancel_l _ _ ((c*b)* ((c*a)/(c*b)))).
+ rewrite <- div_mod.
+ rewrite div_mul_cancel_l; auto.
+ rewrite <- mul_assoc, <- mul_add_distr_l, mul_cancel_l by order.
+ apply div_mod; order.
+ intro EQ; symmetry in EQ; revert EQ. apply lt_neq, mul_pos_pos; auto.
+Qed.
+
+Lemma mul_mod_distr_r: forall a b c, 0<=a -> 0<b -> 0<c ->
+  (a*c) mod (b*c) == (a mod b) * c.
+Proof.
+ intros.
+ rewrite (mul_comm a c), (mul_comm b c); rewrite mul_mod_distr_l; auto.
+ apply mul_comm.
+Qed.
+
+(** Operations modulo. *)
+
+Theorem mod_mod: forall a n, 0<=a -> 0<n ->
+ (a mod n) mod n == a mod n.
+Proof.
+ intros. destruct (mod_bound a n); auto. rewrite mod_small_iff; auto.
+Qed.
+
+Lemma mul_mod_idemp_l : forall a b n, 0<=a -> 0<=b -> 0<n ->
+ ((a mod n)*b) mod n == (a*b) mod n.
+Proof.
+ intros a b n Ha Hb Hn. symmetry.
+ generalize (mul_nonneg_nonneg _ _ Ha Hb).
+ rewrite (div_mod a n) at 1 2; [|order].
+ rewrite add_comm, (mul_comm n), (mul_comm _ b).
+ rewrite mul_add_distr_l, mul_assoc.
+ intros. rewrite mod_add; auto.
+ rewrite mul_comm; auto.
+ apply mul_nonneg_nonneg; destruct (mod_bound a n); auto.
+Qed.
+
+Lemma mul_mod_idemp_r : forall a b n, 0<=a -> 0<=b -> 0<n ->
+ (a*(b mod n)) mod n == (a*b) mod n.
+Proof.
+ intros. rewrite !(mul_comm a). apply mul_mod_idemp_l; auto.
+Qed.
+
+Theorem mul_mod: forall a b n, 0<=a -> 0<=b -> 0<n ->
+ (a * b) mod n == ((a mod n) * (b mod n)) mod n.
+Proof.
+ intros. rewrite mul_mod_idemp_l, mul_mod_idemp_r; auto.
+ destruct (mod_bound b n); auto.
+Qed.
+
+Lemma add_mod_idemp_l : forall a b n, 0<=a -> 0<=b -> 0<n ->
+ ((a mod n)+b) mod n == (a+b) mod n.
+Proof.
+ intros a b n Ha Hb Hn. symmetry.
+ generalize (add_nonneg_nonneg _ _ Ha Hb).
+ rewrite (div_mod a n) at 1 2; [|order].
+ rewrite <- add_assoc, add_comm, mul_comm.
+ intros. rewrite mod_add; auto.
+ apply add_nonneg_nonneg; auto. destruct (mod_bound a n); auto.
+Qed.
+
+Lemma add_mod_idemp_r : forall a b n, 0<=a -> 0<=b -> 0<n ->
+ (a+(b mod n)) mod n == (a+b) mod n.
+Proof.
+ intros. rewrite !(add_comm a). apply add_mod_idemp_l; auto.
+Qed.
+
+Theorem add_mod: forall a b n, 0<=a -> 0<=b -> 0<n ->
+ (a+b) mod n == (a mod n + b mod n) mod n.
+Proof.
+ intros. rewrite add_mod_idemp_l, add_mod_idemp_r; auto.
+ destruct (mod_bound b n); auto.
+Qed.
+
+Lemma div_div : forall a b c, 0<=a -> 0<b -> 0<c ->
+ (a/b)/c == a/(b*c).
+Proof.
+ intros a b c Ha Hb Hc.
+ apply div_unique with (b*((a/b) mod c) + a mod b); auto.
+ (* begin 0<= ... <b*c *)
+ destruct (mod_bound (a/b) c), (mod_bound a b); auto using div_pos.
+ split.
+ apply add_nonneg_nonneg; auto.
+ apply mul_nonneg_nonneg; order.
+ apply lt_le_trans with (b*((a/b) mod c) + b).
+ rewrite <- add_lt_mono_l; auto.
+ rewrite <- mul_succ_r, <- mul_le_mono_pos_l, le_succ_l; auto.
+ (* end 0<= ... < b*c *)
+ rewrite (div_mod a b) at 1; [|order].
+ rewrite add_assoc, add_cancel_r.
+ rewrite <- mul_assoc, <- mul_add_distr_l, mul_cancel_l by order.
+ apply div_mod; order.
+Qed.
+
+(** A last inequality: *)
+
+Theorem div_mul_le:
+ forall a b c, 0<=a -> 0<b -> 0<=c -> c*(a/b) <= (c*a)/b.
+Proof.
+ intros.
+ apply div_le_lower_bound; auto.
+ apply mul_nonneg_nonneg; auto.
+ rewrite mul_assoc, (mul_comm b c), <- mul_assoc.
+ apply mul_le_mono_nonneg_l; auto.
+ apply mul_div_le; auto.
+Qed.
+
+(** mod is related to divisibility *)
+
+Lemma mod_divides : forall a b, 0<=a -> 0<b ->
+ (a mod b == 0 <-> exists c, a == b*c).
+Proof.
+ split.
+ intros. exists (a/b). rewrite div_exact; auto.
+ intros (c,Hc). symmetry; apply mod_unique with c; auto.
+  split; order.
+  nzsimpl; auto.
+Qed.
+
+End NZDivPropFunct.
+
diff --git a/theories/Numbers/NatInt/NZMulOrder.v b/theories/Numbers/NatInt/NZMulOrder.v
index 306b69022c..b6e4849f5e 100644
--- a/theories/Numbers/NatInt/NZMulOrder.v
+++ b/theories/Numbers/NatInt/NZMulOrder.v
@@ -202,6 +202,11 @@ Proof.
 intros; rewrite mul_comm; now apply mul_pos_neg.
 Qed.
 
+Theorem mul_nonneg_nonneg : forall n m, 0 <= n -> 0 <= m -> 0 <= n*m.
+Proof.
+intros. rewrite <- (mul_0_l m). apply mul_le_mono_nonneg; order.
+Qed.
+
 Theorem lt_1_mul_pos : forall n m, 1 < n -> 0 < m -> 1 < n * m.
 Proof.
 intros n m H1 H2. apply -> (mul_lt_mono_pos_r m) in H1.
diff --git a/theories/Numbers/Natural/Abstract/NDiv.v b/theories/Numbers/Natural/Abstract/NDiv.v
new file mode 100644
index 0000000000..9ff9c08cf2
--- /dev/null
+++ b/theories/Numbers/Natural/Abstract/NDiv.v
@@ -0,0 +1,249 @@
+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+
+(** Euclidean Division *)
+
+Require Import NAxioms NProperties NZDiv.
+
+Open Scope NumScope.
+
+Module Type NDiv (Import N : NAxiomsSig).
+
+ Parameter Inline div : t -> t -> t.
+ Parameter Inline modulo : t -> t -> t.
+
+ Infix "/" := div : NumScope.
+ Infix "mod" := modulo (at level 40, no associativity) : NumScope.
+
+ Instance div_wd : Proper (eq==>eq==>eq) div.
+ Instance mod_wd : Proper (eq==>eq==>eq) modulo.
+
+ Axiom div_mod : forall a b, b ~= 0 -> a == b*(a/b) + (a mod b).
+ Axiom mod_upper_bound : forall a b, b ~= 0 -> a mod b < b.
+
+End NDiv.
+
+Module Type NDivSig := NAxiomsSig <+ NDiv.
+
+Module NDivPropFunct (Import N : NDivSig).
+ Module Import NP := NPropFunct N.
+
+(** We benefit from what already exists for NZ *)
+
+ Module N' <: NZDivSig.
+  Include N.
+  Lemma mod_bound : forall a b, 0<=a -> 0<b -> 0 <= a mod b < b.
+  Proof. split. apply le_0_l. apply mod_upper_bound. order. Qed.
+ End N'.
+ Module Import NZDivP := NZDivPropFunct N'.
+
+ Ltac auto' := try rewrite <- neq_0_lt_0; auto using le_0_l.
+
+(** Let's now state again theorems, but without useless hypothesis. *)
+
+(** Uniqueness theorems *)
+
+Theorem div_mod_unique :
+ forall b q1 q2 r1 r2, r1<b -> r2<b ->
+  b*q1+r1 == b*q2+r2 -> q1 == q2 /\ r1 == r2.
+Proof. intros. apply div_mod_unique with b; auto'. Qed.
+
+Theorem div_unique:
+ forall a b q r, r<b -> a == b*q + r -> q == a/b.
+Proof. intros; apply div_unique with r; auto'. Qed.
+
+Theorem mod_unique:
+ forall a b q r, r<b -> a == b*q + r -> r == a mod b.
+Proof. intros. apply mod_unique with q; auto'. Qed.
+
+(** A division by itself returns 1 *)
+
+Lemma div_same : forall a, a~=0 -> a/a == 1.
+Proof. intros. apply div_same; auto'. Qed.
+
+Lemma mod_same : forall a, a~=0 -> a mod a == 0.
+Proof. intros. apply mod_same; auto'. Qed.
+
+(** A division of a small number by a bigger one yields zero. *)
+
+Theorem div_small: forall a b, a<b -> a/b == 0.
+Proof. intros. apply div_small; auto'. Qed.
+
+(** Same situation, in term of modulo: *)
+
+Theorem mod_small: forall a b, a<b -> a mod b == a.
+Proof. intros. apply mod_small; auto'. Qed.
+
+(** * Basic values of divisions and modulo. *)
+
+Lemma div_0_l: forall a, a~=0 -> 0/a == 0.
+Proof. intros. apply div_0_l; auto'. Qed.
+
+Lemma mod_0_l: forall a, a~=0 -> 0 mod a == 0.
+Proof. intros. apply mod_0_l; auto'. Qed.
+
+Lemma div_1_r: forall a, a/1 == a.
+Proof. intros. apply div_1_r; auto'. Qed.
+
+Lemma mod_1_r: forall a, a mod 1 == 0.
+Proof. intros. apply mod_1_r; auto'. Qed.
+
+Lemma div_1_l: forall a, 1<a -> 1/a == 0.
+Proof. exact div_1_l. Qed.
+
+Lemma mod_1_l: forall a, 1<a -> 1 mod a == 1.
+Proof. exact mod_1_l. Qed.
+
+Lemma div_mul : forall a b, b~=0 -> (a*b)/b == a.
+Proof. intros. apply div_mul; auto'. Qed.
+
+Lemma mod_mul : forall a b, b~=0 -> (a*b) mod b == 0.
+Proof. intros. apply mod_mul; auto'. Qed.
+
+
+(** * Order results about mod and div *)
+
+(** A modulo cannot grow beyond its starting point. *)
+
+Theorem mod_le: forall a b, b~=0 -> a mod b <= a.
+Proof. intros. apply mod_le; auto'. Qed.
+
+Lemma div_str_pos : forall a b, 0<b<=a -> 0 < a/b.
+Proof. exact div_str_pos. Qed.
+
+Lemma div_small_iff : forall a b, b~=0 -> (a/b==0 <-> a<b).
+Proof. intros. apply div_small_iff; auto'. Qed.
+
+Lemma mod_small_iff : forall a b, b~=0 -> (a mod b == a <-> a<b).
+Proof. intros. apply mod_small_iff; auto'. Qed.
+
+Lemma div_str_pos_iff : forall a b, b~=0 -> (0<a/b <-> b<=a).
+Proof. intros. apply div_str_pos_iff; auto'. Qed.
+
+
+(** As soon as the divisor is strictly greater than 1,
+    the division is strictly decreasing. *)
+
+Lemma div_lt : forall a b, 0<a -> 1<b -> a/b < a.
+Proof. exact div_lt. Qed.
+
+(** [le] is compatible with a positive division. *)
+
+Lemma div_le_mono : forall a b c, c~=0 -> a<=b -> a/c <= b/c.
+Proof. intros. apply div_le_mono; auto'. Qed.
+
+Lemma mul_div_le : forall a b, b~=0 -> b*(a/b) <= a.
+Proof. intros. apply mul_div_le; auto'. Qed.
+
+Lemma mul_succ_div_gt: forall a b, b~=0 -> a < b*(S (a/b)).
+Proof. intros; apply mul_succ_div_gt; auto'. Qed.
+
+(** The previous inequality is exact iff the modulo is zero. *)
+
+Lemma div_exact : forall a b, b~=0 -> (a == b*(a/b) <-> a mod b == 0).
+Proof. intros. apply div_exact; auto'. Qed.
+
+(** Some additionnal inequalities about div. *)
+
+Theorem div_lt_upper_bound:
+  forall a b q, b~=0 -> a < b*q -> a/b < q.
+Proof. intros. apply div_lt_upper_bound; auto'. Qed.
+
+Theorem div_le_upper_bound:
+  forall a b q, b~=0 -> a <= b*q -> a/b <= q.
+Proof. intros; apply div_le_upper_bound; auto'. Qed.
+
+Theorem div_le_lower_bound:
+  forall a b q, b~=0 -> b*q <= a -> q <= a/b.
+Proof. intros; apply div_le_lower_bound; auto'. Qed.
+
+(** A division respects opposite monotonicity for the divisor *)
+
+Lemma div_le_compat_l: forall p q r, 0<q<=r -> p/r <= p/q.
+Proof. intros. apply div_le_compat_l. auto'. auto. Qed.
+
+(** * Relations between usual operations and mod and div *)
+
+Lemma mod_add : forall a b c, c~=0 ->
+ (a + b * c) mod c == a mod c.
+Proof. intros. apply mod_add; auto'. Qed.
+
+Lemma div_add : forall a b c, c~=0 ->
+ (a + b * c) / c == a / c + b.
+Proof. intros. apply div_add; auto'. Qed.
+
+Lemma div_add_l: forall a b c, b~=0 ->
+ (a * b + c) / b == a + c / b.
+Proof. intros. apply div_add_l; auto'. Qed.
+
+(** Cancellations. *)
+
+Lemma div_mul_cancel_r : forall a b c, b~=0 -> c~=0 ->
+ (a*c)/(b*c) == a/b.
+Proof. intros. apply div_mul_cancel_r; auto'. Qed.
+
+Lemma div_mul_cancel_l : forall a b c, b~=0 -> c~=0 ->
+ (c*a)/(c*b) == a/b.
+Proof. intros. apply div_mul_cancel_l; auto'. Qed.
+
+Lemma mul_mod_distr_r: forall a b c, b~=0 -> c~=0 ->
+  (a*c) mod (b*c) == (a mod b) * c.
+Proof. intros. apply mul_mod_distr_r; auto'. Qed.
+
+Lemma mul_mod_distr_l: forall a b c, b~=0 -> c~=0 ->
+  (c*a) mod (c*b) == c * (a mod b).
+Proof. intros. apply mul_mod_distr_l; auto'. Qed.
+
+(** Operations modulo. *)
+
+Theorem mod_mod: forall a n, n~=0 ->
+ (a mod n) mod n == a mod n.
+Proof. intros. apply mod_mod; auto'. Qed.
+
+Lemma mul_mod_idemp_l : forall a b n, n~=0 ->
+ ((a mod n)*b) mod n == (a*b) mod n.
+Proof. intros. apply mul_mod_idemp_l; auto'. Qed.
+
+Lemma mul_mod_idemp_r : forall a b n, n~=0 ->
+ (a*(b mod n)) mod n == (a*b) mod n.
+Proof. intros. apply mul_mod_idemp_r; auto'. Qed.
+
+Theorem mul_mod: forall a b n, n~=0 ->
+ (a * b) mod n == ((a mod n) * (b mod n)) mod n.
+Proof. intros. apply mul_mod; auto'. Qed.
+
+Lemma add_mod_idemp_l : forall a b n, n~=0 ->
+ ((a mod n)+b) mod n == (a+b) mod n.
+Proof. intros. apply add_mod_idemp_l; auto'. Qed.
+
+Lemma add_mod_idemp_r : forall a b n, n~=0 ->
+ (a+(b mod n)) mod n == (a+b) mod n.
+Proof. intros. apply add_mod_idemp_r; auto'. Qed.
+
+Theorem add_mod: forall a b n, n~=0 ->
+ (a+b) mod n == (a mod n + b mod n) mod n.
+Proof. intros. apply add_mod; auto'. Qed.
+
+Lemma div_div : forall a b c, b~=0 -> c~=0 ->
+ (a/b)/c == a/(b*c).
+Proof. intros. apply div_div; auto'. Qed.
+
+(** A last inequality: *)
+
+Theorem div_mul_le:
+ forall a b c, b~=0 -> c*(a/b) <= (c*a)/b.
+Proof. intros. apply div_mul_le; auto'. Qed.
+
+(** mod is related to divisibility *)
+
+Lemma mod_divides : forall a b, b~=0 ->
+ (a mod b == 0 <-> exists c, a == b*c).
+Proof. intros. apply mod_divides; auto'. Qed.
+
+End NDivPropFunct.
+
diff --git a/theories/Numbers/Natural/Peano/NPeano.v b/theories/Numbers/Natural/Peano/NPeano.v
index 8974ef1143..97ac9f8729 100644
--- a/theories/Numbers/Natural/Peano/NPeano.v
+++ b/theories/Numbers/Natural/Peano/NPeano.v
@@ -10,10 +10,7 @@
 
 (*i $Id$ i*)
 
-Require Import Arith.
-Require Import Min.
-Require Import Max.
-Require Import NAxioms NProperties.
+Require Import Arith MinMax NAxioms NProperties.
 
 (** * Implementation of [NAxiomsSig] by [nat] *)
 
@@ -164,7 +161,75 @@ Definition max := max.
 
 End NPeanoAxiomsMod.
 
-(* Now we apply the largest property functor *)
+(** Now we apply the largest property functor *)
 
 Module Export NPeanoPropMod := NPropFunct NPeanoAxiomsMod.
 
+
+
+(** Euclidean Division *)
+
+Definition divF div x y := if leb y x then S (div (x-y) y) else 0.
+Definition modF mod x y := if leb y x then mod (x-y) y else x.
+Definition initF (_ _ : nat) := 0.
+
+Fixpoint loop {A} (F:A->A)(i:A) (n:nat) : A :=
+ match n with
+  | 0 => i
+  | S n => F (loop F i n)
+ end.
+
+Definition div x y := loop divF initF x x y.
+Definition modulo x y := loop modF initF x x y.
+Infix "/" := div : nat_scope.
+Infix "mod" := modulo (at level 40, no associativity) : nat_scope.
+
+Lemma div_mod : forall x y, y<>0 -> x = y*(x/y) + x mod y.
+Proof.
+ cut (forall n x y, y<>0 -> x<=n ->
+       x = y*(loop divF initF n x y) + (loop modF initF n x y)).
+ intros H x y Hy. apply H; auto.
+ induction n.
+ simpl; unfold initF; simpl. intros. nzsimpl. auto with arith.
+ simpl; unfold divF at 1, modF at 1.
+ intros.
+ destruct (leb y x) as [ ]_eqn:L;
+  [apply leb_complete in L | apply leb_complete_conv in L].
+ rewrite mul_succ_r, <- add_assoc, (add_comm y), add_assoc.
+ rewrite <- IHn; auto.
+ symmetry; apply sub_add; auto.
+ rewrite <- NPeanoAxiomsMod.lt_succ_r.
+ apply lt_le_trans with x; auto.
+ apply lt_minus; auto. rewrite <- neq_0_lt_0; auto.
+ nzsimpl; auto.
+Qed.
+
+Lemma mod_upper_bound : forall x y, y<>0 -> x mod y < y.
+Proof.
+ cut (forall n x y, y<>0 -> x<=n -> loop modF initF n x y < y).
+ intros H x y Hy. apply H; auto.
+ induction n.
+ simpl; unfold initF. intros. rewrite <- neq_0_lt_0; auto.
+ simpl; unfold modF at 1.
+ intros.
+ destruct (leb y x) as [ ]_eqn:L;
+  [apply leb_complete in L | apply leb_complete_conv in L]; auto.
+ apply IHn; auto.
+ rewrite <- NPeanoAxiomsMod.lt_succ_r.
+ apply lt_le_trans with x; auto.
+ apply lt_minus; auto. rewrite <- neq_0_lt_0; auto.
+Qed.
+
+Require Import NDiv.
+
+Module NDivMod <: NDivSig.
+ Include NPeanoAxiomsMod.
+ Definition div := div.
+ Definition modulo := modulo.
+ Definition div_mod := div_mod.
+ Definition mod_upper_bound := mod_upper_bound.
+ Program Instance div_wd : Proper (eq==>eq==>eq) div.
+ Program Instance mod_wd : Proper (eq==>eq==>eq) modulo.
+End NDivMod.
+
+Module Export NDivPropMod := NDivPropFunct NDivMod.
diff --git a/theories/Numbers/vo.itarget b/theories/Numbers/vo.itarget
index 003f20f2d6..7c29450eab 100644
--- a/theories/Numbers/vo.itarget
+++ b/theories/Numbers/vo.itarget
@@ -22,6 +22,9 @@ Integer/Abstract/ZLt.vo
 Integer/Abstract/ZMulOrder.vo
 Integer/Abstract/ZMul.vo
 Integer/Abstract/ZProperties.vo
+Integer/Abstract/ZDivCoq.vo
+Integer/Abstract/ZDivOcaml.vo
+Integer/Abstract/ZDivMath.vo
 Integer/BigZ/BigZ.vo
 Integer/BigZ/ZMake.vo
 Integer/Binary/ZBinary.vo
@@ -38,6 +41,7 @@ NatInt/NZMul.vo
 NatInt/NZOrder.vo
 NatInt/NZProperties.vo
 NatInt/NZDomain.vo
+NatInt/NZDiv.vo
 Natural/Abstract/NAddOrder.vo
 Natural/Abstract/NAdd.vo
 Natural/Abstract/NAxioms.vo
@@ -49,6 +53,7 @@ Natural/Abstract/NOrder.vo
 Natural/Abstract/NStrongRec.vo
 Natural/Abstract/NSub.vo
 Natural/Abstract/NProperties.vo
+Natural/Abstract/NDiv.vo
 Natural/BigN/BigN.vo
 Natural/BigN/Nbasic.vo
 Natural/BigN/NMake.vo
diff --git a/theories/ZArith/ZOdiv.v b/theories/ZArith/ZOdiv.v
index 28b664aa48..c73673a858 100644
--- a/theories/ZArith/ZOdiv.v
+++ b/theories/ZArith/ZOdiv.v
@@ -7,7 +7,8 @@
 (************************************************************************)
 
 
-Require Import BinPos BinNat Nnat ZArith_base ROmega ZArithRing.
+Require Import BinPos BinNat Nnat ZArith_base ROmega ZArithRing
+ ZBinary ZDivOcaml Morphisms.
 Require Export ZOdiv_def.
 Require Zdiv.
 
@@ -243,6 +244,29 @@ Proof.
   unfold ZOmod, ZOdiv_eucl; destruct Pdiv_eucl; simpl; auto with zarith.
 Qed.
 
+(** We know enough to prove that [ZOdiv] and [ZOmod] are instances of
+    one of the abstract Euclidean divisions of Numbers. *) 
+
+Module ZODiv <: ZDiv ZBinAxiomsMod.
+ Definition div := ZOdiv.
+ Definition modulo := ZOmod.
+ Program Instance div_wd : Proper (eq==>eq==>eq) div.
+ Program Instance mod_wd : Proper (eq==>eq==>eq) modulo.
+
+ Definition div_mod := fun a b (_:b<>0) => ZO_div_mod_eq a b.
+ Definition mod_bound := ZOmod_lt_pos_pos.
+ Definition mod_opp_l := fun a b (_:b<>0) => ZOmod_opp_l a b.
+ Definition mod_opp_r := fun a b (_:b<>0) => ZOmod_opp_r a b.
+End ZODiv.
+
+Module ZODivMod := ZBinAxiomsMod <+ ZODiv.
+
+(** We hence benefit from generic results about this abstract division. *)
+
+Module Z.
+ Include ZDivPropFunct ZODivMod.
+End Z.
+
 (** * Unicity results *)
 
 Definition Remainder a b r :=
@@ -291,10 +315,7 @@ Qed.
 Theorem ZOdiv_unique:
  forall a b q r, 0 <= a -> 0 <= r < b ->
    a = b*q + r -> q = a/b.
-Proof.
-  intros; eapply ZOdiv_unique_full; eauto.
-  red; romega with *.
-Qed.
+Proof. exact Z.div_unique. Qed.
 
 Theorem ZOmod_unique_full:
  forall a b q r, Remainder a b r ->
@@ -306,10 +327,7 @@ Qed.
 Theorem ZOmod_unique:
  forall a b q r, 0 <= a -> 0 <= r < b ->
    a = b*q + r -> r = a mod b.
-Proof.
-  intros; eapply ZOmod_unique_full; eauto.
-  red; romega with *.
-Qed.
+Proof. exact Z.mod_unique. Qed.
 
 (** * Basic values of divisions and modulo. *)
 
@@ -334,56 +352,37 @@ Proof.
 Qed.
 
 Lemma ZOmod_1_r: forall a, a mod 1 = 0.
-Proof.
-  intros; symmetry; apply ZOmod_unique_full with a; auto with zarith.
-  rewrite Remainder_equiv; red; simpl; auto with zarith.
-Qed.
+Proof. exact Z.mod_1_r. Qed.
 
 Lemma ZOdiv_1_r: forall a, a/1 = a.
-Proof.
-  intros; symmetry; apply ZOdiv_unique_full with 0; auto with zarith.
-  rewrite Remainder_equiv; red; simpl; auto with zarith.
-Qed.
+Proof. exact Z.div_1_r. Qed.
 
 Hint Resolve ZOmod_0_l ZOmod_0_r ZOdiv_0_l ZOdiv_0_r ZOdiv_1_r ZOmod_1_r
  : zarith.
 
 Lemma ZOdiv_1_l: forall a, 1 < a -> 1/a = 0.
-Proof.
-  intros; symmetry; apply ZOdiv_unique with 1; auto with zarith.
-Qed.
+Proof. exact Z.div_1_l. Qed.
 
 Lemma ZOmod_1_l: forall a, 1 < a ->  1 mod a = 1.
-Proof.
-  intros; symmetry; apply ZOmod_unique with 0; auto with zarith.
-Qed.
+Proof. exact Z.mod_1_l. Qed.
 
 Lemma ZO_div_same : forall a:Z, a<>0 -> a/a = 1.
-Proof.
-  intros; symmetry; apply ZOdiv_unique_full with 0; auto with *.
-  rewrite Remainder_equiv; red; simpl; romega with *.
-Qed.
+Proof. exact Z.div_same. Qed.
 
 Lemma ZO_mod_same : forall a, a mod a = 0.
 Proof.
-  destruct a; intros; symmetry.
-  compute; auto.
-  apply ZOmod_unique with 1; auto with *; romega with *.
-  apply ZOmod_unique_full with 1; auto with *; red; romega with *.
+ intros. destruct (Z_eq_dec a 0); subst. apply ZOmod_0_l.
+ apply Z.mod_same; auto.
 Qed.
 
 Lemma ZO_mod_mult : forall a b, (a*b) mod b = 0.
 Proof.
-  intros a b; destruct (Z_eq_dec b 0) as [Hb|Hb].
-  subst; simpl; rewrite ZOmod_0_r; auto with zarith.
-  symmetry; apply ZOmod_unique_full with a; [ red; romega with * | ring ].
+ intros. destruct (Z_eq_dec b 0); subst. rewrite Zmult_0_r. apply ZOmod_0_l.
+ apply Z.mod_mul; auto.
 Qed.
 
 Lemma ZO_div_mult : forall a b:Z, b <> 0 -> (a*b)/b = a.
-Proof.
-  intros; symmetry; apply ZOdiv_unique_full with 0; auto with zarith;
-   [ red; romega with * | ring].
-Qed.
+Proof. exact Z.div_mul. Qed.
 
 (** * Order results about ZOmod and ZOdiv *)
 
@@ -391,16 +390,8 @@ Qed.
 
 Lemma ZO_div_pos: forall a b, 0 <= a -> 0 <= b -> 0 <= a/b.
 Proof.
-  intros.
-  destruct (Zle_lt_or_eq 0 b H0).
-  assert (H2:=ZOmod_lt_pos_pos a b H H1).
-  rewrite (ZO_div_mod_eq a b) in H.
-  destruct (Z_lt_le_dec (a/b) 0); auto.
-  assert (b*(a/b) <= -b).
-   replace (-b) with (b*-1); [ | ring].
-   apply Zmult_le_compat_l; auto with zarith.
-  romega.
-  subst b; rewrite ZOdiv_0_r; auto.
+ intros. destruct (Z_eq_dec b 0); subst. rewrite ZOdiv_0_r; auto.
+ apply Z.div_pos; auto with zarith.
 Qed.
 
 (** As soon as the divisor is greater or equal than 2,
@@ -408,97 +399,41 @@ Qed.
 
 Lemma ZO_div_lt : forall a b:Z, 0 < a -> 2 <= b -> a/b < a.
 Proof.
-  intros.
-  assert (Hb : 0 < b) by romega.
-  assert (H1 : 0 <= a/b) by (apply ZO_div_pos; auto with zarith).
-  assert (H2 : 0 <= a mod b < b) by (apply ZOmod_lt_pos_pos; auto with zarith).
-  destruct (Zle_lt_or_eq 0 (a/b) H1) as [H3|H3]; [ | rewrite <- H3; auto].
-  pattern a at 2; rewrite (ZO_div_mod_eq a b).
-  apply Zlt_le_trans with (2*(a/b)).
-  romega.
-  apply Zle_trans with (b*(a/b)).
-  apply Zmult_le_compat_r; auto.
-  romega.
+ intros. apply Z.div_lt; auto with zarith.
 Qed.
 
 (** A division of a small number by a bigger one yields zero. *)
 
 Theorem ZOdiv_small: forall a b, 0 <= a < b -> a/b = 0.
-Proof.
-  intros a b H; apply sym_equal; apply ZOdiv_unique with a; auto with zarith.
-Qed.
+Proof. exact Z.div_small. Qed.
 
 (** Same situation, in term of modulo: *)
 
 Theorem ZOmod_small: forall a n, 0 <= a < n -> a mod n = a.
-Proof.
-  intros a b H; apply sym_equal; apply ZOmod_unique with 0; auto with zarith.
-Qed.
+Proof. exact Z.mod_small. Qed.
 
 (** [Zge] is compatible with a positive division. *)
 
-Lemma ZO_div_monotone_pos : forall a b c:Z, 0<=c -> 0<=a<=b -> a/c <= b/c.
-Proof.
-  intros.
-  destruct H0.
-  destruct (Zle_lt_or_eq 0 c H);
-   [ clear H | subst c; do 2 rewrite ZOdiv_0_r; auto].
-  generalize (ZO_div_mod_eq a c).
-  generalize (ZOmod_lt_pos_pos a c H0 H2).
-  generalize (ZO_div_mod_eq b c).
-  generalize (ZOmod_lt_pos_pos b c (Zle_trans _ _ _ H0 H1) H2).
-  intros.
-  elim (Z_le_gt_dec (a / c) (b / c)); auto with zarith.
-  intro.
-  absurd (a - b >= 1).
-  omega.
-  replace (a-b) with (c * (a/c-b/c) + a mod c - b mod c) by
-    (symmetry; pattern a at 1; rewrite H5; pattern b at 1; rewrite H3; ring).
-  assert (c * (a / c - b / c) >= c * 1).
-  apply Zmult_ge_compat_l.
-  omega.
-  omega.
-  assert (c * 1 = c).
-  ring.
-  omega.
-Qed.
-
 Lemma ZO_div_monotone : forall a b c, 0<=c -> a<=b -> a/c <= b/c.
-Proof.
-  intros.
-  destruct (Z_le_gt_dec 0 a).
-  apply ZO_div_monotone_pos; auto with zarith.
-  destruct (Z_le_gt_dec 0 b).
-  apply Zle_trans with 0.
-  apply Zle_left_rev.
-  simpl.
-  rewrite <- ZOdiv_opp_l.
-  apply ZO_div_pos; auto with zarith.
-  apply ZO_div_pos; auto with zarith.
-  rewrite <-(Zopp_involutive a), (ZOdiv_opp_l (-a)).
-  rewrite <-(Zopp_involutive b), (ZOdiv_opp_l (-b)).
-  generalize (ZO_div_monotone_pos (-b) (-a) c H).
-  romega.
+Proof. 
+ intros. destruct (Z_eq_dec c 0); subst. rewrite !ZOdiv_0_r; auto.
+ apply Z.div_le_mono; auto with zarith.
 Qed.
 
+(** Compatitility: *)
+Lemma ZO_div_monotone_pos : forall a b c:Z, 0<=c -> 0<=a<=b -> a/c <= b/c.
+Proof. intros; apply ZO_div_monotone; intuition. Qed.
+
 (** With our choice of division, rounding of (a/b) is always done toward zero: *)
 
 Lemma ZO_mult_div_le : forall a b:Z, 0 <= a -> 0 <= b*(a/b) <= a.
 Proof.
-  intros a b Ha.
-  destruct b as [ |b|b].
-  simpl; auto with zarith.
-  split.
-  apply Zmult_le_0_compat; auto with zarith.
-  apply ZO_div_pos; auto with zarith.
-  generalize (ZO_div_mod_eq a (Zpos b)) (ZOmod_lt_pos_pos a (Zpos b) Ha); romega with *.
-  change (Zneg b) with (-Zpos b); rewrite ZOdiv_opp_r, Zmult_opp_opp.
-  split.
-  apply Zmult_le_0_compat; auto with zarith.
-  apply ZO_div_pos; auto with zarith.
-  generalize (ZO_div_mod_eq a (Zpos b)) (ZOmod_lt_pos_pos a (Zpos b) Ha); romega with *.
+ intros. destruct (Z_eq_dec b 0); subst. rewrite !ZOdiv_0_r; auto with zarith.
+ apply Z.mul_div_le; auto with zarith.
 Qed.
 
+(** TODO: finish adapting to generic results *)
+
 Lemma ZO_mult_div_ge : forall a b:Z, a <= 0 -> a <= b*(a/b) <= 0.
 Proof.
   intros a b Ha.