(* 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 Monoid_kernel.
Require Export Free_monoid.
Section Generated_monoid_def.
Variable M : MONOID.
Variable A : part_set M.
Definition generated_monoid : submonoid M :=
image_monoid_hom (FM_lift (inj_part A)).
End Generated_monoid_def.

Lemma generated_monoid_minimal :
forall (M : MONOID) (A : part_set M) (H : submonoid M),
included A H -> included (generated_monoid A) H.