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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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(* 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                                                     *)
(** * lin_dependence.v *)
Set Implicit Arguments.
Unset Strict Implicit.
Section MAIN.
From Algebra Require Export Union.
Require Export subspaces.
Require Export cast_between_subsets.
Require Export mult_by_scalars.
Require Export sums.
Require Export seq_set_seq.
Require Export distinct.
Require Export const.
Section defs.
Variable F : field.
Variable V : vectorspace F.
(** %\tocdef{lin\_{\em[in]}dep}% *)
Definition lin_dep (X : part_set V) :=
exists n : Nat,
(exists a : seq (S n) F,
(exists v : seq (S n) X,
distinct v /\
~ a =' const_seq (S n) (zero F) in _ /\
sum (mult_by_scalars a (Map_embed v)) =' (zero V) in _)).
Definition lin_indep (X : part_set V) := ~ lin_dep X.
Definition lin_indep' (X : part_set V) :=
forall (n : Nat) (a : seq (S n) F) (v : seq (S n) X),
distinct v ->
~ a =' const_seq (S n) (zero F) in _ ->
~ sum (mult_by_scalars a (Map_embed v)) =' (zero V) in _.

Lemma lin_dep_comp :
forall X Y : part_set V, X =' Y in _ -> lin_dep X -> lin_dep Y.