English
Let M be a commutative monoid. For n ∈ ℕ, f : Fin (n + 1) → M, a,b ∈ Fin n, the product over i ∈ Ioo a.castSucc b.castSucc equals the product over i ∈ Ioo a b of f(i.castSucc).
Русский
Пусть M — коммутативный моноид. Для n ∈ ℕ, f : Fin(n+1) → M и a,b ∈ Fin n, произведение по i ∈ Ioo a.castSucc b.castSucc равно произведению по i ∈ Ioo a b от f i.castSucc.
LaTeX
$$$$ \\displaystyle \\prod_{i \\in \\mathrm{Ioo}(a^{\\mathrm{castSucc}}, b^{\\mathrm{castSucc}})} f(i) = \\prod_{i \\in \\mathrm{Ioo} a b} f(i^{\\mathrm{castSucc}}) $$$$
Lean4
@[to_additive]
theorem prod_Iic_castAdd (m : ℕ) (f : Fin (n + m) → M) (a : Fin n) :
∏ i ≤ a.castAdd m, f i = ∏ i ≤ a, f (i.castAdd m) := by simp [← map_castAddEmb_Iic]
