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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.             *)
(*                                                                    *)
(* 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                                                     *)
(** * bases.v *)
Set Implicit Arguments.
Unset Strict Implicit.
Require Export lin_dependence.
Require Export spans.
Require Export random_facts.
Section MAIN.
Variable F : field.
Variable V : vectorspace F.
Section Defs.
(** %\tocdef{is\_basis}% *)
Definition is_basis (X : part_set V) : Prop :=
generates X (full V) /\ lin_indep X.

Lemma is_basis_comp :
forall X Y : part_set V, X =' Y in _ -> is_basis X -> is_basis Y.