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.

(* Specifications: *)
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 _.