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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+
+
+include "nat/orders.ma".
+include "group.ma".
+
+let rec gpow (G : abelian_group) (x : G) (n : nat) on n : G ≝
+  match n with [ O ⇒ 0 | S m ⇒ x + gpow ? x m].
+  
+interpretation "additive abelian group pow" 'times n x =
+  (cic:/matita/divisible_group/gpow.con _ x n).
+  
+record dgroup : Type ≝ {
+  dg_carr:> abelian_group;
+  dg_prop: ∀x:dg_carr.∀n:nat.∃y.S n * y ≈ x
+}.
+
+lemma divide: ∀G:dgroup.G → nat → G.
+intros (G x n); cases (dg_prop G x n); apply w; 
+qed.
+
+interpretation "divisible group divide" 'divide x n =
+  (cic:/matita/divisible_group/divide.con _ x n).
+
+lemma divide_divides: 
+  ∀G:dgroup.∀x:G.∀n. S n * (x / n) ≈ x.
+intro G; cases G; unfold divide; intros (x n); simplify;
+cases (f x n); simplify; exact H;
+qed.  
+  
+lemma feq_mul: ∀G:dgroup.∀x,y:G.∀n.x≈y → n * x ≈ n * y.
+intros (G x y n H); elim n; [apply eq_reflexive]
+simplify; apply (Eq≈ (x + (n1 * y)) H1);
+apply (Eq≈ (y+n1*y) H (eq_reflexive ??));
+qed.
+
+lemma div1: ∀G:dgroup.∀x:G.x/O ≈ x.
+intro G; cases G; unfold divide; intros; simplify;
+cases (f x O); simplify; simplify in H; intro; apply H;
+apply (Ap≪ ? (plus_comm ???));
+apply (Ap≪ w (zero_neutral ??)); assumption;
+qed.
+
+lemma apmul_ap: ∀G:dgroup.∀x,y:G.∀n.S n * x # S n * y → x # y.
+intros 4 (G x y n); elim n; [2:
+  simplify in a;
+  cases (applus ????? a); [assumption]
+  apply f; assumption;]
+apply (plus_cancr_ap ??? 0); assumption;
+qed.
+
+lemma plusmul: ∀G:dgroup.∀x,y:G.∀n.n * (x+y) ≈ n * x + n * y.
+intros (G x y n); elim n; [
+  simplify; apply (Eq≈ 0 ? (zero_neutral ? 0)); apply eq_reflexive]
+simplify; apply eq_sym; apply (Eq≈ (x+y+(n1*x+n1*y))); [
+  apply (Eq≈ (x+(n1*x+(y+(n1*y))))); [
+    apply eq_sym; apply plus_assoc;]
+  apply (Eq≈ (x+((n1*x+y+(n1*y))))); [
+    apply feq_plusl; apply plus_assoc;]
+  apply (Eq≈ (x+(y+n1*x+n1*y))); [
+    apply feq_plusl; apply feq_plusr; apply plus_comm;] 
+  apply (Eq≈ (x+(y+(n1*x+n1*y)))); [
+    apply feq_plusl; apply eq_sym; apply plus_assoc;]
+  apply plus_assoc;]
+apply feq_plusl; apply eq_sym; assumption;
+qed. 
+
+lemma mulzero: ∀G:dgroup.∀n.n*0 ≈ 0. [2: apply dg_carr; apply G]
+intros; elim n; [simplify; apply eq_reflexive]
+simplify; apply (Eq≈ ? (zero_neutral ??)); assumption; 
+qed.
+
+let rec gpowS (G : abelian_group) (x : G) (n : nat) on n : G ≝
+  match n with [ O ⇒ x | S m ⇒ gpowS ? x m + x].
+  
+lemma gpowS_gpow: ∀G:dgroup.∀e:G.∀n. S n * e ≈ gpowS ? e n.
+intros (G e n); elim n; simplify; [
+  apply (Eq≈ ? (plus_comm ???));apply zero_neutral]
+apply (Eq≈ ?? (plus_comm ???));  
+apply (Eq≈ (e+S n1*e) ? H); clear H; simplify; apply eq_reflexive;
+qed.
+
+lemma divpow: ∀G:dgroup.∀e:G.∀n. e ≈ gpowS ? (e/n) n.
+intros (G e n); apply (Eq≈ ?? (gpowS_gpow ?(e/n) n));
+apply eq_sym; apply divide_divides;
+qed.