star_sum

theorem star_sum : ∀ {R : Type u_1} [inst : AddCommMonoid R] [inst_1 : StarAddMonoid R] {α : Type u_2} (s : Finset α) (f : α → R), star (s.sum fun x => f x) = s.sum fun x => star (f x)

The theorem `star_sum` states that for any type `R` equipped with both an additive commutative monoid structure and a `StarAddMonoid` structure, and for any finite set `s` of elements of an arbitrary type `α`, the "star" of the sum of the function `f` applied to all elements in `s` is equal to the sum of the "star" of `f` applied to each individual element in `s`. Here, "star" is a unary operation from the `StarAddMonoid` structure on `R` and the sum is computed over the set `s` using the `Finset.sum` function. In mathematical terms, this is similar to saying $\star\left(\sum_{x\in s} f(x)\right) = \sum_{x\in s} \star(f(x))$ for a finite set $s$ and a function $f$ from $s$ to $R$.

More concisely: For any type `R` with additive commutative monoid and `StarAddMonoid` structures, and finite set `s` of elements from an arbitrary type `α`, the "star" of the sum of `f(x)` for all `x` in `s` equals the sum of the "stars" of `f(x)` for all `x` in `s`. In symbols, $\star\left(\sum\_{x\in s} f(x)\right) = \sum\_{x\in s} \star(f(x))$.