Lemmas about products and sums over finite sets in Option α
#
In this file we prove formulas for products and sums over Finset.insertNone s
and
Finset.eraseNone s
.
theorem
Finset.add_sum_eq_sum_insertNone
{α : Type u_1}
{M : Type u_2}
[AddCommMonoid M]
(f : α → M)
(x : M)
(s : Finset α)
:
theorem
Finset.mul_prod_eq_prod_insertNone
{α : Type u_1}
{M : Type u_2}
[CommMonoid M]
(f : α → M)
(x : M)
(s : Finset α)
:
theorem
Finset.sum_eraseNone
{α : Type u_1}
{M : Type u_2}
[AddCommMonoid M]
(f : α → M)
(s : Finset (Option α))
:
((Finset.eraseNone s).sum fun (x : α) => f x) = s.sum fun (x : Option α) => Option.elim' 0 f x
theorem
Finset.prod_eraseNone
{α : Type u_1}
{M : Type u_2}
[CommMonoid M]
(f : α → M)
(s : Finset (Option α))
:
((Finset.eraseNone s).prod fun (x : α) => f x) = s.prod fun (x : Option α) => Option.elim' 1 f x