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+(************************************************************************)
+(*  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.
+
+Module Type NDivSpecific (Import N : NAxiomsSig')(Import DM : DivMod' N).
+ Axiom mod_upper_bound : forall a b, b ~= 0 -> a mod b < b.
+End NDivSpecific.
+
+Module Type NDivSig := NAxiomsSig <+ DivMod <+ NZDivCommon <+ NDivSpecific.
+Module Type NDivSig' := NAxiomsSig' <+ DivMod' <+ NZDivCommon <+ NDivSpecific.
+
+Module NDivPropFunct (Import N : NDivSig')(Import NP : NPropSig N).
+
+(** We benefit from what already exists for NZ *)
+
+ Module ND <: NZDiv N.
+  Definition div := div.
+  Definition modulo := modulo.
+  Definition div_wd := div_wd.
+  Definition mod_wd := mod_wd.
+  Definition div_mod := div_mod.
+  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 ND.
+ Module Import NZDivP := NZDivPropFunct N NP ND.
+
+ 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.
+