English
In a commutative group with zero, the Möbius inversion identity holds on a well-behaved set s with the extra nonzero hypotheses (hf, hg) ensuring the expressions are meaningful.
Русский
В группе с нулём тождество Моебиуса верно на допустимом множестве s с дополнительными условиями не нулевых значений f,g.
LaTeX
$$$\forall f,g : \mathbb{N} \to R\ (s:Set\mathbb{N}) (hs : ∀ m n, m \mid n → n ∈ s → m ∈ s) (hf : ∀ n > 0, f n ≠ 0) (hg : ∀ n > 0, g n ≠ 0) : (∀ n > 0, n ∈ s → (\prod i ∈ n.divisors, f i) = g n) \\iff \\ (∀ n > 0, n ∈ s → (\prod x ∈ n.divisorsAntidiagonal, g x.snd ^ μ x.fst) = f n).$$
Lean4
theorem bernoulli'_spec' (n : ℕ) :
(∑ k ∈ antidiagonal n, ((k.1 + k.2).choose k.2 : ℚ) / (k.2 + 1) * bernoulli' k.1) = 1 :=
by
refine ((sum_antidiagonal_eq_sum_range_succ_mk _ n).trans ?_).trans (bernoulli'_spec n)
refine sum_congr rfl fun x hx => ?_
simp only [add_tsub_cancel_of_le, mem_range_succ_iff.mp hx, cast_sub]
