calc
  _ = ∑ d in P.divisors,
        ∑ a in Fintype.piFinset fun _i : Fin k => P.divisors,
          if ∏ i, a i = d ∧ d ∣ P then if ↑d ≤ x then (d : ℝ)⁻¹ else 0 else 0 := ?_
  _ = ∑ a in Fintype.piFinset fun _i : Fin k => P.divisors,
        if ∏ i, a i ∣ P then if ↑(∏ i, a i) ≤ x then ∏ i, (a i : ℝ)⁻¹ else 0 else 0 := ?_
  _ ≤ ∑ a in Fintype.piFinset fun _i : Fin k => P.divisors,
        if ↑(∏ i, a i) ≤ x then ∏ i, (a i : ℝ)⁻¹ else 0 := ?_ _ ≤ ∑ a in Fintype.piFinset fun _i : Fin k => P.divisors,
        ∏ i, if ↑(a i) ≤ x then (a i : ℝ)⁻¹ else 0 := ?_
  _ = ∏ _i : Fin k, ∑ d in P.divisors, if ↑d ≤ x then (d : ℝ)⁻¹ else 0 := by rw [prod_univ_sum]
  _ = (∑ d in P.divisors, if ↑d ≤ x then (d : ℝ)⁻¹ else 0) ^ k := by rw [prod_const, card_fin]
  _ ≤ (1 + Real.log x) ^ k := ?_
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
⊢ (∑ d ∈ P.divisors, if ↑d ≤ x then ↑k ^ ω d / ↑d else 0) ≤ (1 + x.log) ^ k
case calc_1
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
⊢ (∑ d ∈ P.divisors, if ↑d ≤ x then ↑k ^ ω d / ↑d else 0) =
    ∑ d ∈ P.divisors,
      ∑ a ∈ Fintype.piFinset fun _i => P.divisors,
        if ∏ i : Fin k, a i = d ∧ d ∣ P then if ↑d ≤ x then (↑d)⁻¹ else 0 else 0

case calc_2
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
⊢ (∑ d ∈ P.divisors,
      ∑ a ∈ Fintype.piFinset fun _i => P.divisors,
        if ∏ i : Fin k, a i = d ∧ d ∣ P then if ↑d ≤ x then (↑d)⁻¹ else 0 else 0) =
    ∑ a ∈ Fintype.piFinset fun _i => P.divisors,
      if ∏ i : Fin k, a i ∣ P then if ↑(∏ i : Fin k, a i) ≤ x then ∏ i : Fin k, (↑(a i))⁻¹ else 0 else 0

case calc_3
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
⊢ (∑ a ∈ Fintype.piFinset fun _i => P.divisors,
      if ∏ i : Fin k, a i ∣ P then if ↑(∏ i : Fin k, a i) ≤ x then ∏ i : Fin k, (↑(a i))⁻¹ else 0 else 0) ≤
    ∑ a ∈ Fintype.piFinset fun _i => P.divisors, if ↑(∏ i : Fin k, a i) ≤ x then ∏ i : Fin k, (↑(a i))⁻¹ else 0

case calc_4
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
⊢ (∑ a ∈ Fintype.piFinset fun _i => P.divisors, if ↑(∏ i : Fin k, a i) ≤ x then ∏ i : Fin k, (↑(a i))⁻¹ else 0) ≤
    ∑ a ∈ Fintype.piFinset fun _i => P.divisors, ∏ i : Fin k, if ↑(a i) ≤ x then (↑(a i))⁻¹ else 0

case calc_5
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
⊢ (∑ d ∈ P.divisors, if ↑d ≤ x then (↑d)⁻¹ else 0) ^ k ≤ (1 + x.log) ^ k