(* 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                                                     *)
(** * lin_combinations.v *)
Set Implicit Arguments.
Unset Strict Implicit.
Require Export distinct.
Require Export distribution_lemmas.
Require Export sums2.
Require Export omit_facts.
Require Export cast_between_subsets.
Section lin_dep_def.
(** %\tocdef{is\_lin\_comb}% *)
Definition is_lin_comb (F : field) (V : vectorspace F)
(x : V) (S : part_set V) :=
exists n : Nat,
(exists a : seq n F,
(exists v : seq n S, x =' sum (mult_by_scalars a (Map_embed v)) in _)).

(* Specifications: *)
Lemma is_lin_comb_comp :
forall (F : field) (V : vectorspace F) (x y : V) (S T : part_set V),
S =' T in _ -> x =' y in _ -> is_lin_comb x S -> is_lin_comb y T.