Section Distributivity.
Variable R : Type.
Variables zero one : R.
Local Notation "0" := zero.
Local Notation "1" := one.
Variable times : Monoid.mul_law 0.
Local Notation "*%M" := times (at level 0).
Local Notation "x * y" := (times x y).
Variable plus : Monoid.add_law 0 *%M.
Local Notation "+%M" := plus (at level 0).
Local Notation "x + y" := (plus x y).

Lemma big_distrl I r a (P : pred I) F :
\big[+%M/0]_(i <- r | P i) F i * a = \big[+%M/0]_(i <- r | P i) (F i * a).