(* 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_comb_facts.v *)
Set Implicit Arguments.
Unset Strict Implicit.
Require Export algebraic_span_facts.
Require Export seq_set_seq.
(** - Miscellaneous facts about linear combinations:  - Suppose $x$ is a linear combination of $W$, and all $w\in W$ are linear combinations of $W'$, then $x$ is a linear combination of $W'$ *)

Lemma lin_comb_casting :
forall (F : field) (V : vectorspace F) (W W' : part_set V) (x : V),
is_lin_comb x W ->
(forall w : W, is_lin_comb (subtype_elt w) W') -> is_lin_comb x W'.