English
Let I be a type and f: I → Type be a family of AddMonoids, each equipped with a distributive action of α. Then the function type ∀ i ∈ I, f(i) carries a distributive action of α defined pointwise by (a, x) ↦ (i ↦ a • x(i)).
Русский
Пусть I — множество индексов, a_i — семейство видов f(i) с маркеровыми добавочными моноидами, на которые действует α распределённо. Тогда функция-тип ∀ i f(i) несёт распределимое действие α, заданное покомпонентно: (a • x)(i) = a • x(i).
LaTeX
$$$\text{If } \forall i, f(i) \text{ is an AddMonoid with a DistribMulAction by } \alpha, \text{ then } (\forall i, f(i)) \text{ has DistribMulAction by } \alpha,$ with $(a \cdot x)(i) = a \cdot x(i)$.$$
Lean4
instance distribMulAction (α) {m : Monoid α} {n : ∀ i, AddMonoid <| f i} [∀ i, DistribMulAction α <| f i] :
@DistribMulAction α (∀ i : I, f i) m (@Pi.addMonoid I f n) :=
{ Pi.mulAction _, Pi.distribSMul _ with }
