English
Let M be a commutative monoid. For n,m ∈ ℕ, f : Fin (n + m) → M, and a,b ∈ Fin n, the product over i ∈ Ico (a.castSucc m) (b.castSucc m) equals the product over i ∈ Ico a b of f(i.castSucc m).
Русский
Пусть M — коммутативный моноид. Для n,m ∈ ℕ, f : Fin(n+m) → M и a,b ∈ Fin n, произведение по i ∈ Ico(a.castSucc m)(b.castSucc m) равно произведению по i ∈ Ico a b от f(i.castSucc m).
LaTeX
$$$$ \\displaystyle \\prod_{i \\in \\mathrm{Ico}(a^{\\mathrm{castSucc} m}, b^{\\mathrm{castSucc} m})} f(i) = \\prod_{i \\in \\mathrm{Ico} a b} f(i^{\\mathrm{castSucc} m}) $$$$
Lean4
@[to_additive]
theorem prod_Ico_castAdd (m : ℕ) (f : Fin (n + m) → M) (a b : Fin n) :
∏ i ∈ Ico (a.castAdd m) (b.castAdd m), f i = ∏ i ∈ Ico a b, f (i.castAdd m) := by simp [← map_castAddEmb_Ico]
