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(* 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.             *)
(*                                                                    *)
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(* 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.
Unset Standard Proposition Elimination Names.
Require Export Zring.
Require Export Group_kernel.
(** Title "Universal property of integers." *)
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.
End Int_power.
Section Zup1.
Variable G : GROUP.
Variable r : G.
Fixpoint nat_to_group (n : nat) : G :=
match n with
| O => monoid_unit G
| S n' => sgroup_law G (nat_to_group n') r
end.
Definition pos_abs : forall x : Z, (x > 0)%Z -> positive.
intros x; try assumption.
case x.
intros H'; red in H'.
simpl in H'.
inversion H'.
intros p H'.
exact p.
intros p H'; red in H'.
simpl in H'.
inversion H'.
Defined.

Lemma pos_abs_ok : forall (x : Z) (px : (x > 0)%Z), x = Zpos (pos_abs px).
intros x; try assumption.