(* This program is free software; you can redistribute it and/or      *)
(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1   *)
(* of the License, or (at your option) any later version.             *)
(*                                                                    *)
(* This program is distributed in the hope that it will be useful,    *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of     *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the      *)
(* GNU General Public License for more details.                       *)
(*                                                                    *)
(* You should have received a copy of the GNU Lesser General Public   *)
(* License along with this program; if not, write to the Free         *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA                                                     *)
Set Automatic Coercions Import.
Set Implicit Arguments.
Unset Strict Implicit.
Require Export Zring.
Section Int_power.
Variable G : GROUP.
Set Strict Implicit.
Unset Implicit Arguments.
Definition group_square (x : G) : G := sgroup_law G x x.
Set Implicit Arguments.
Unset Strict Implicit.
Fixpoint group_power_pos (g : G) (p : positive) {struct p} : G :=
match p with
| xH => g
| xO p' => group_square (group_power_pos g p')
| xI p' => sgroup_law G (group_square (group_power_pos g p')) g
end.
Set Strict Implicit.
Unset Implicit Arguments.
Definition group_power (g : G) (z : ZZ) : G :=
match z with
| Z0 => monoid_unit G
| Zpos p => group_power_pos g p
| Zneg p => group_power_pos (group_inverse G g) p
end.
Set Implicit Arguments.
Unset Strict Implicit.
End Int_power.
Section Lemmas.
Variable G : GROUP.

Lemma group_power_zero :
forall g : G, Equal (group_power G g (monoid_unit ZZ)) (monoid_unit G).