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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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(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the      *)
(* GNU General Public License for more details.                       *)
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(* License along with this program; if not, write to the Free         *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA                                                     *)
(** %\subsection*{ support :  distribution\_lemmas.v }%*)
Set Implicit Arguments.
Unset Strict Implicit.
From Algebra Require Field_facts.
Export Sets Categories Parts Sgroup_cat Monoid_cat Group_cat Abelian_group_cat Ring_cat.
Export Field_facts.
From Algebra Require Ring_facts.
Export Ring_facts.
From Algebra Require Export Field_cat.
Require Export mult_by_scalars.
Require Export const.
Require Export sums.
(** - $r\sum_i a_i = \sum_i ra_i$ (in rings), - $r\sum_i v_i = \sum_i rv_i$ (in modules) and - $\sum_i r_i(r'_iv_i) = \sum_i (r_ir'_i)v_i$ *)

Lemma RING_sum_mult_dist_l :
forall (R : ring) (n : Nat) (r : R) (a : seq n R),
r rX sum a =' sum (pointwise (uncurry (RING_comp (R:=R))) (const_seq n r) a)
in _.