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(* GNU General Public License for more details.                       *)
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(* 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 - $\sum_i r_i(r'_iv_i) = \sum_i (r_ir'_i)v_i$ *)
simple induction n.
simpl in |- *.
auto with algebra.
intros.
apply Trans with (r rX sum (hdtl a)); auto with algebra; unfold hdtl in |- *.
apply Trans with (r rX (head a +' sum (Seqtl a))); auto with algebra.
apply Trans with (r rX head a +' r rX sum (Seqtl a)); auto with algebra.
apply
Trans
with
(r rX head a +'
sum
(pointwise (uncurry (RING_comp (R:=R))) (Seqtl (const_seq (S n0) r))
(Seqtl a))).
apply SGROUP_comp; auto with algebra.
apply
Trans
with
(sum (pointwise (uncurry (RING_comp (R:=R))) (const_seq n0 r) (Seqtl a)));
auto with algebra.
apply sum_comp.
apply toMap.
apply pointwise_comp; auto with algebra.
apply Sym.
change (Seqtl (const_seq (S n0) r) =' const_seq (pred (S n0)) r in _) in |- *.
apply Seqtl_const_seq.
apply Sym.
apply
Trans
with
(sum
(hdtl (pointwise (uncurry (RING_comp (R:=R))) (const_seq (S n0) r) a)));
auto with algebra.
unfold hdtl in |- *.
apply
Trans
with
(head (pointwise (uncurry (RING_comp (R:=R))) (const_seq (S n0) r) a) +'
sum
(pointwise (uncurry (RING_comp (R:=R))) (Seqtl (const_seq (S n0) r))
(Seqtl a))); auto with algebra.
apply
Trans
with
(head (pointwise (uncurry (RING_comp (R:=R))) (const_seq (S n0) r) a) +'
sum
(Seqtl (pointwise (uncurry (RING_comp (R:=R))) (const_seq (S n0) r) a)));
auto with algebra.
Qed.

Lemma MODULE_sum_mult_dist_l :
forall (R : ring) (M : module R) (n : Nat) (r : R) (a : seq n M),
r mX sum a =' sum (mult_by_scalars (const_seq n r) a) in _.