(* 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 Implicit Arguments.
Unset Strict Implicit.
Require Export Sub_group.
Require Export Abelian_group_cat.
(** Title "Tools for building groups." *)
Section Group.
Variable E : Setoid.
Variable genlaw : E -> E -> E.
Variable e : E.
Variable geninv : E -> E.
Hypothesis
fcomp :
forall x x' y y' : E,
Equal x x' -> Equal y y' -> Equal (genlaw x y) (genlaw x' y').
Hypothesis
genlawassoc :
forall x y z : E, Equal (genlaw (genlaw x y) z) (genlaw x (genlaw y z)).
Hypothesis eunitgenlawr : forall x : E, Equal (genlaw x e) x.
Hypothesis invcomp : forall x y : E, Equal x y -> Equal (geninv x) (geninv y).
Hypothesis geninvr : forall x : E, Equal (genlaw x (geninv x)) e.

(* Specifications: *)
Lemma geninvl : forall x : E, Equal (genlaw (geninv x) x) e.
intros x; try assumption.