apply sum_congr rfl
case pos.a
s : SelbergSieve
d : ℕ
h_dvd : d ∣ P
⊢ (∑ m ∈ P.divisors, if d * m ∣ P then S⁻¹ * ↑(μ d) * if d ∣ d * m ∧ ↑(d * m) ^ 2 ≤ y then g (d * m) else 0 else 0) =
    ∑ i ∈ P.divisors, ν d * ((ν d)⁻¹ * g d * ↑(μ d) * S⁻¹ * if (↑d * ↑i) ^ 2 ≤ y ∧ i.Coprime d then g i else 0)

case pos.a
s : SelbergSieve
d : ℕ
h_dvd : d ∣ P
⊢ ∀ (l : ℕ), l ∣ P → ¬d ∣ l → (S⁻¹ * ↑(μ d) * if d ∣ l ∧ ↑l ^ 2 ≤ y then g l else 0) = 0
case pos.a
s : SelbergSieve
d : ℕ
h_dvd : d ∣ P
⊢ ∀ x ∈ P.divisors,
    (if d * x ∣ P then S⁻¹ * ↑(μ d) * if d ∣ d * x ∧ ↑(d * x) ^ 2 ≤ y then g (d * x) else 0 else 0) =
      ν d * ((ν d)⁻¹ * g d * ↑(μ d) * S⁻¹ * if (↑d * ↑x) ^ 2 ≤ y ∧ x.Coprime d then g x else 0)

case pos.a
s : SelbergSieve
d : ℕ
h_dvd : d ∣ P
⊢ ∀ (l : ℕ), l ∣ P → ¬d ∣ l → (S⁻¹ * ↑(μ d) * if d ∣ l ∧ ↑l ^ 2 ≤ y then g l else 0) = 0