awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (hypothesis)
Goal:
ι : Type u_1
R : Type u_2
inst : CommSemiring R
f : ι → ArithmeticFunction R
hf : ∀ (i : ι), IsMultiplicative.{u_2} (R := R) (f i)
hf' : Multipliable f
h_1 : ∀ (s : Finset ι), IsMultiplicative.{u_2} (R := R) (∏ b ∈ s, f b)
key :
  ∀ (n : ℕ),
    ∀ᶠ (i : Finset ι) in (SummationFilter.unconditional ι).filter,
      Eq.{u_2 + 1} (α := R) ((∏ b ∈ i, f b : (a : ℕ) → R) n : R) ((∏' (b : ι), f b : (a : ℕ) → R) n : R)
m n : ℕ
hmn : m.Coprime n
h :
  ∀ (s : Finset ι),
    (∏ b ∈ s, f b : (a : ℕ) → R) (m * n) =
      HMul.hMul (α := R) (β := R) ((∏ b ∈ s, f b : (a : ℕ) → R) m : R) ((∏ b ∈ s, f b : (a : ℕ) → R) n : R)
⊢ (eulerProduct f : (a : ℕ) → R) (m * n) =
    HMul.hMul (α := R) (β := R) ((eulerProduct f : (a : ℕ) → R) m : R) ((eulerProduct f : (a : ℕ) → R) n : R)
have h2 : ∀ᶠ (i : Finset ι) in (SummationFilter.unconditional ι).filter,
  Eq.{u_2 + 1} (α := R) ((∏ b ∈ i, f b : (a : ℕ) → R) (m * n) : R) ((∏' (b : ι), f b : (a : ℕ) → R) (m * n) : R)
Tactic: have h2 := key (m * n)
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (hypothesis)
Goal:
ι : Type u_1
R : Type u_2
inst : CommSemiring R
f : ι → ArithmeticFunction R
hf : ∀ (i : ι), IsMultiplicative.{u_2} (R := R) (f i)
hf' : Multipliable f
h_1 : ∀ (s : Finset ι), IsMultiplicative.{u_2} (R := R) (∏ b ∈ s, f b)
key :
  ∀ (n : ℕ),
    ∀ᶠ (i : Finset ι) in (SummationFilter.unconditional ι).filter,
      Eq.{u_2 + 1} (α := R) ((∏ b ∈ i, f b : (a : ℕ) → R) n : R) ((∏' (b : ι), f b : (a : ℕ) → R) n : R)
m n : ℕ
hmn : m.Coprime n
h :
  ∀ (s : Finset ι),
    (∏ b ∈ s, f b : (a : ℕ) → R) (m * n) =
      HMul.hMul (α := R) (β := R) ((∏ b ∈ s, f b : (a : ℕ) → R) m : R) ((∏ b ∈ s, f b : (a : ℕ) → R) n : R)
h2 :
  ∀ᶠ (i : Finset ι) in (SummationFilter.unconditional ι).filter,
    Eq.{u_2 + 1} (α := R) ((∏ b ∈ i, f b : (a : ℕ) → R) (m * n) : R) ((∏' (b : ι), f b : (a : ℕ) → R) (m * n) : R)
⊢ (eulerProduct f : (a : ℕ) → R) (m * n) =
    HMul.hMul (α := R) (β := R) ((eulerProduct f : (a : ℕ) → R) m : R) ((eulerProduct f : (a : ℕ) → R) n : R)
have h2₁ : ∀ᶠ (i : Finset ι) in (SummationFilter.unconditional ι).filter,
  HMul.hMul (α := R) (β := R) ((∏ b ∈ i, f b : (a : ℕ) → R) m : R) ((∏ b ∈ i, f b : (a : ℕ) → R) n : R) =
    (∏' (b : ι), f b : (a : ℕ) → R) (m * n)
Tactic: simp_rw [h] at h2
Cleared hypotheses: h2
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (goal)
Goal:
ι : Type u_1
R : Type u_2
inst : CommSemiring R
f : ι → ArithmeticFunction R
hf : ∀ (i : ι), IsMultiplicative.{u_2} (R := R) (f i)
hf' : ¬Multipliable f
⊢ IsMultiplicative.{u_2} (R := R) (eulerProduct f)
have neg : IsMultiplicative.{u_2} (R := R) 1
Tactic: rw [eulerProduct, tprod_eq_one_of_not_multipliable hf']
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (goal)
Goal:
ι : Type u_1
R : Type u_2
inst : CommSemiring R
q : ι → ℕ
hq : Northcott q
f : ι → PowerSeries R
hf : ∀ (i : ι), Eq (α := R) ((PowerSeries.constantCoeff : (a : PowerSeries R) → R) (f i) : R) (1 : R)
n : ℕ
⊢ ∀ᶠ (s : Finset ι) in atTop,
    Eq.{u_2 + 1} (α := R)
      ((∏ i ∈ s, (ofPowerSeries (q i) : (a : PowerSeries R) → ArithmeticFunction R) (f i) : (a : ℕ) → R) n : R)
      ((eulerProduct fun (i : ι) ↦ (ofPowerSeries (q i) : (a : PowerSeries R) → ArithmeticFunction R) (f i) :
            (a : ℕ) → R)
          n :
        R)
have hf₁ : ∀ (n : ℕ),
  ∀ᶠ (i : ι) in cofinite,
    Eq (α := R)
      ((((ofPowerSeries (q i) : (a : PowerSeries R) → ArithmeticFunction R) (f i) : ArithmeticFunction R) : (a : ℕ) → R)
          n :
        R)
      ((1 : (a : ℕ) → R) n : R)
Tactic: apply tendsTo_eulerProduct_of_tendsTo
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (goal)
Goal:
ι : Type u_1
R : Type u_2
inst : CommSemiring R
q : ι → ℕ
hq : Northcott q
f : ι → PowerSeries R
hf : ∀ (i : ι), Eq (α := R) ((PowerSeries.constantCoeff : (a : PowerSeries R) → R) (f i) : R) (1 : R)
n : ℕ
⊢ ∀ (n : ℕ),
    ∀ᶠ (i : ι) in cofinite,
      Eq (α := R)
        ((((ofPowerSeries (q i) : (a : PowerSeries R) → ArithmeticFunction R) (f i) : ArithmeticFunction R) :
              (a : ℕ) → R)
            n :
          R)
        ((1 : (a : ℕ) → R) n : R)
have hf₁ : ∀ (n : ℕ) (i : ι) (hi : n + 1 ≤ q i),
  Eq (α := R)
    ((((ofPowerSeries (q i) : (a : PowerSeries R) → ArithmeticFunction R) (f i) : ArithmeticFunction R) : (a : ℕ) → R)
        n :
      R)
    ((1 : (a : ℕ) → R) n : R)
Tactic: refine fun n ↦ (tendsto_atTop.mp ((northcott_iff_tendsto q).mp hq) (n + 1)).mono fun i hi ↦ ?_
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (goal)
Goal:
ι : Type u_1
R : Type u_2
inst : CommSemiring R
q : ι → ℕ
hq : Northcott q
f : ι → PowerSeries R
hf : ∀ (i : ι), Eq (α := R) ((PowerSeries.constantCoeff : (a : PowerSeries R) → R) (f i) : R) (1 : R)
n_1 n : ℕ
i : ι
hi : n + 1 ≤ q i
⊢ Eq (α := R)
    ((((ofPowerSeries (q i) : (a : PowerSeries R) → ArithmeticFunction R) (f i) : ArithmeticFunction R) : (a : ℕ) → R)
        n :
      R)
    ((1 : (a : ℕ) → R) n : R)
have hf₁ : ∀ (hi : HAdd.hAdd (α := ℕ) 0 1 ≤ q i),
  Eq (α := R)
    ((((ofPowerSeries (q i) : (a : PowerSeries R) → ArithmeticFunction R) (f i) : ArithmeticFunction R) : (a : ℕ) → R)
        0 :
      R)
    ((1 : (a : ℕ) → R) 0 : R)
Tactic: rcases n with rfl | rfl | n
Cleared hypotheses: n, hi
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (goal)
Goal:
ι : Type u_1
R : Type u_2
inst : CommSemiring R
q : ι → ℕ
hq : Northcott q
f : ι → PowerSeries R
hf : ∀ (i : ι), Eq (α := R) ((PowerSeries.constantCoeff : (a : PowerSeries R) → R) (f i) : R) (1 : R)
n_1 n : ℕ
i : ι
hi : n + 1 ≤ q i
hf₁ :
  ∀ (hi : HAdd.hAdd (α := ℕ) 0 1 ≤ q i),
    Eq (α := R)
      ((((ofPowerSeries (q i) : (a : PowerSeries R) → ArithmeticFunction R) (f i) : ArithmeticFunction R) : (a : ℕ) → R)
          0 :
        R)
      ((1 : (a : ℕ) → R) 0 : R)
⊢ Eq (α := R)
    ((((ofPowerSeries (q i) : (a : PowerSeries R) → ArithmeticFunction R) (f i) : ArithmeticFunction R) : (a : ℕ) → R)
        n :
      R)
    ((1 : (a : ℕ) → R) n : R)
have hf₂ : ∀ (hi : HAdd.hAdd (α := ℕ) 0 1 + 1 ≤ q i),
  Eq (α := R)
    ((((ofPowerSeries (q i) : (a : PowerSeries R) → ArithmeticFunction R) (f i) : ArithmeticFunction R) : (a : ℕ) → R)
        (HAdd.hAdd (α := ℕ) 0 1) :
      R)
    ((1 : (a : ℕ) → R) (HAdd.hAdd (α := ℕ) 0 1) : R)
Tactic: rcases n with rfl | rfl | n
Cleared hypotheses: n, hi
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (goal)
Goal:
ι : Type u_1
R : Type u_2
inst : CommSemiring R
q : ι → ℕ
hq : Northcott q
f : ι → PowerSeries R
hf : ∀ (i : ι), Eq (α := R) ((PowerSeries.constantCoeff : (a : PowerSeries R) → R) (f i) : R) (1 : R)
n_1 n : ℕ
i : ι
hi : n + 1 ≤ q i
hf₁ :
  ∀ (hi : HAdd.hAdd (α := ℕ) 0 1 ≤ q i),
    Eq (α := R)
      ((((ofPowerSeries (q i) : (a : PowerSeries R) → ArithmeticFunction R) (f i) : ArithmeticFunction R) : (a : ℕ) → R)
          0 :
        R)
      ((1 : (a : ℕ) → R) 0 : R)
hf₂ :
  ∀ (hi : HAdd.hAdd (α := ℕ) 0 1 + 1 ≤ q i),
    Eq (α := R)
      ((((ofPowerSeries (q i) : (a : PowerSeries R) → ArithmeticFunction R) (f i) : ArithmeticFunction R) : (a : ℕ) → R)
          (HAdd.hAdd (α := ℕ) 0 1) :
        R)
      ((1 : (a : ℕ) → R) (HAdd.hAdd (α := ℕ) 0 1) : R)
⊢ Eq (α := R)
    ((((ofPowerSeries (q i) : (a : PowerSeries R) → ArithmeticFunction R) (f i) : ArithmeticFunction R) : (a : ℕ) → R)
        n :
      R)
    ((1 : (a : ℕ) → R) n : R)
have hf₃ : ∀ (n : ℕ) (hi : n + 1 + 1 + 1 ≤ q i),
  Eq (α := R)
    ((((ofPowerSeries (q i) : (a : PowerSeries R) → ArithmeticFunction R) (f i) : ArithmeticFunction R) : (a : ℕ) → R)
        (n + 1 + 1) :
      R)
    ((1 : (a : ℕ) → R) (n + 1 + 1) : R)
Tactic: rcases n with rfl | rfl | n
Cleared hypotheses: n, hi
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (hypothesis)
Goal:
ι : Type u_1
R : Type u_2
inst : CommSemiring R
q : ι → ℕ
hq : Northcott q
f : ι → PowerSeries R
hf : ∀ (i : ι), Eq (α := R) ((PowerSeries.constantCoeff : (a : PowerSeries R) → R) (f i) : R) (1 : R)
n_1 : ℕ
i : ι
n : ℕ
hi : n + 1 + 1 + 1 ≤ q i
⊢ Eq (α := R)
    ((((ofPowerSeries (q i) : (a : PowerSeries R) → ArithmeticFunction R) (f i) : ArithmeticFunction R) : (a : ℕ) → R)
        (n + 1 + 1) :
      R)
    ((1 : (a : ℕ) → R) (n + 1 + 1) : R)
have hqi : 1 < q i
Tactic: have hqi : 1 < q i := by lia
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (goal)
Goal:
ι : Type u_1
R : Type u_2
inst : CommSemiring R
q : ι → ℕ
hq : Northcott q
f : ι → PowerSeries R
hf : ∀ (i : ι), Eq (α := R) ((PowerSeries.constantCoeff : (a : PowerSeries R) → R) (f i) : R) (1 : R)
n_1 : ℕ
i : ι
n : ℕ
hi : n + 1 + 1 + 1 ≤ q i
hqi : 1 < q i
⊢ Eq (α := R)
    ((((ofPowerSeries (q i) : (a : PowerSeries R) → ArithmeticFunction R) (f i) : ArithmeticFunction R) : (a : ℕ) → R)
        (n + 1 + 1) :
      R)
    ((1 : (a : ℕ) → R) (n + 1 + 1) : R)
have hf₁ : ¬∃ (a : ℕ), q i ^ a = n + 1 + 1
Tactic: rw [ofPowerSeries_apply hqi, Function.extend_apply', Pi.zero_apply, one_apply_ne (by lia)]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (hypothesis)
Goal:
ι : Type u_1
R : Type u_2
inst : CommSemiring R
q : ι → ℕ
hq : Northcott q
f : ι → PowerSeries R
hf : ∀ (i : ι), Eq (α := R) ((PowerSeries.constantCoeff : (a : PowerSeries R) → R) (f i) : R) (1 : R)
n_1 : ℕ
i : ι
n : ℕ
hi : n + 1 + 1 + 1 ≤ q i
hqi : 1 < q i
k : ℕ
hk : q i ^ k = n + 1 + 1
⊢ False
have h : k ≠ 0
Tactic: have h : k ≠ 0 := fun h ↦ by simp_all
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Liouville (goal)
Goal:
m n : ℕ
⊢ (liouville : (a : ℕ) → ℤ) (m * n) =
    HMul.hMul (α := ℤ) (β := ℤ) ((liouville : (a : ℕ) → ℤ) m : ℤ) ((liouville : (a : ℕ) → ℤ) n : ℤ)
have pos : ∀ (hm : m = 0),
  (liouville : (a : ℕ) → ℤ) (m * n) =
    HMul.hMul (α := ℤ) (β := ℤ) ((liouville : (a : ℕ) → ℤ) m : ℤ) ((liouville : (a : ℕ) → ℤ) n : ℤ)
Tactic: by_cases hm : m = 0
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Liouville (goal)
Goal:
m n : ℕ
pos :
  ∀ (hm : m = 0),
    (liouville : (a : ℕ) → ℤ) (m * n) =
      HMul.hMul (α := ℤ) (β := ℤ) ((liouville : (a : ℕ) → ℤ) m : ℤ) ((liouville : (a : ℕ) → ℤ) n : ℤ)
⊢ (liouville : (a : ℕ) → ℤ) (m * n) =
    HMul.hMul (α := ℤ) (β := ℤ) ((liouville : (a : ℕ) → ℤ) m : ℤ) ((liouville : (a : ℕ) → ℤ) n : ℤ)
have neg : ∀ (hm : ¬m = 0),
  (liouville : (a : ℕ) → ℤ) (m * n) =
    HMul.hMul (α := ℤ) (β := ℤ) ((liouville : (a : ℕ) → ℤ) m : ℤ) ((liouville : (a : ℕ) → ℤ) n : ℤ)
Tactic: by_cases hm : m = 0
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Liouville (goal)
Goal:
m n : ℕ
hm : ¬m = 0
⊢ (liouville : (a : ℕ) → ℤ) (m * n) =
    HMul.hMul (α := ℤ) (β := ℤ) ((liouville : (a : ℕ) → ℤ) m : ℤ) ((liouville : (a : ℕ) → ℤ) n : ℤ)
have pos : ∀ (hn : n = 0),
  (liouville : (a : ℕ) → ℤ) (m * n) =
    HMul.hMul (α := ℤ) (β := ℤ) ((liouville : (a : ℕ) → ℤ) m : ℤ) ((liouville : (a : ℕ) → ℤ) n : ℤ)
Tactic: by_cases hn : n = 0
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Liouville (goal)
Goal:
m n : ℕ
hm : ¬m = 0
pos :
  ∀ (hn : n = 0),
    (liouville : (a : ℕ) → ℤ) (m * n) =
      HMul.hMul (α := ℤ) (β := ℤ) ((liouville : (a : ℕ) → ℤ) m : ℤ) ((liouville : (a : ℕ) → ℤ) n : ℤ)
⊢ (liouville : (a : ℕ) → ℤ) (m * n) =
    HMul.hMul (α := ℤ) (β := ℤ) ((liouville : (a : ℕ) → ℤ) m : ℤ) ((liouville : (a : ℕ) → ℤ) n : ℤ)
have neg : ∀ (hn : ¬n = 0),
  (liouville : (a : ℕ) → ℤ) (m * n) =
    HMul.hMul (α := ℤ) (β := ℤ) ((liouville : (a : ℕ) → ℤ) m : ℤ) ((liouville : (a : ℕ) → ℤ) n : ℤ)
Tactic: by_cases hn : n = 0
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Misc (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
f : ℕ → R
⊢ IsMultiplicative (ArithmeticFunction.prodPrimeFactors f)
have rw : Eq (α := R) ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) 1 : R) (1 : R) ∧
  ∀ {m n : ℕ},
    m ≠ 0 →
      n ≠ 0 →
        m.Coprime n →
          (ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) (m * n) =
            HMul.hMul (α := R) (β := R) ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) m : R)
              ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) n : R)
Tactic: rw [iff_ne_zero]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Misc (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
f : ℕ → R
⊢ Eq (α := R) ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) 1 : R) (1 : R) ∧
    ∀ {m n : ℕ},
      m ≠ 0 →
        n ≠ 0 →
          m.Coprime n →
            (ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) (m * n) =
              HMul.hMul (α := R) (β := R) ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) m : R)
                ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) n : R)
have simp : ∀ {m n : ℕ},
  ¬m = 0 →
    ¬n = 0 →
      m.Coprime n →
        (ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) (m * n) =
          HMul.hMul (α := R) (β := R) ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) m : R)
            ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) n : R)
Tactic: simp only [ne_eq, one_ne_zero, not_false_eq_true, prodPrimeFactors_apply, primeFactors_one,
    prod_empty, true_and]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Misc (hypothesis)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
f : ℕ → R
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hxy : x.Coprime y
⊢ (ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) (x * y) =
    HMul.hMul (α := R) (β := R) ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) x : R)
      ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) y : R)
have hxy₀ : x * y ≠ 0
Tactic: have hxy₀ : x * y ≠ 0 := mul_ne_zero hx hy
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Misc (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
f : ℕ → R
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hxy : x.Coprime y
hxy₀ : x * y ≠ 0
⊢ (ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) (x * y) =
    HMul.hMul (α := R) (β := R) ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) x : R)
      ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) y : R)
have rw : ∏ p ∈ (x * y).primeFactors, f p =
  HMul.hMul (α := R) (β := R) ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) x : R)
    ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) y : R)
Tactic: rw [prodPrimeFactors_apply hxy₀, prodPrimeFactors_apply hx, prodPrimeFactors_apply hy,
    primeFactors_mul hx hy, ← prod_union hxy.disjoint_primeFactors]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Misc (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
f : ℕ → R
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hxy : x.Coprime y
hxy₀ : x * y ≠ 0
rw :
  ∏ p ∈ (x * y).primeFactors, f p =
    HMul.hMul (α := R) (β := R) ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) x : R)
      ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) y : R)
⊢ (ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) (x * y) =
    HMul.hMul (α := R) (β := R) ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) x : R)
      ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) y : R)
have rw₁ : ∏ p ∈ (x * y).primeFactors, f p =
  HMul.hMul (α := R) (β := R) (∏ p ∈ x.primeFactors, f p : R)
    ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) y : R)
Tactic: rw [prodPrimeFactors_apply hxy₀, prodPrimeFactors_apply hx, prodPrimeFactors_apply hy,
    primeFactors_mul hx hy, ← prod_union hxy.disjoint_primeFactors]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Misc (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
f : ℕ → R
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hxy : x.Coprime y
hxy₀ : x * y ≠ 0
rw :
  ∏ p ∈ (x * y).primeFactors, f p =
    HMul.hMul (α := R) (β := R) ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) x : R)
      ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) y : R)
rw₁ :
  ∏ p ∈ (x * y).primeFactors, f p =
    HMul.hMul (α := R) (β := R) (∏ p ∈ x.primeFactors, f p : R)
      ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) y : R)
⊢ (ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) (x * y) =
    HMul.hMul (α := R) (β := R) ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) x : R)
      ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) y : R)
have rw₂ : ∏ p ∈ (x * y).primeFactors, f p = HMul.hMul (α := R) (∏ p ∈ x.primeFactors, f p) (∏ p ∈ y.primeFactors, f p)
Tactic: rw [prodPrimeFactors_apply hxy₀, prodPrimeFactors_apply hx, prodPrimeFactors_apply hy,
    primeFactors_mul hx hy, ← prod_union hxy.disjoint_primeFactors]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Misc (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
f : ℕ → R
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hxy : x.Coprime y
hxy₀ : x * y ≠ 0
rw :
  ∏ p ∈ (x * y).primeFactors, f p =
    HMul.hMul (α := R) (β := R) ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) x : R)
      ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) y : R)
rw₁ :
  ∏ p ∈ (x * y).primeFactors, f p =
    HMul.hMul (α := R) (β := R) (∏ p ∈ x.primeFactors, f p : R)
      ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) y : R)
rw₂ : ∏ p ∈ (x * y).primeFactors, f p = HMul.hMul (α := R) (∏ p ∈ x.primeFactors, f p) (∏ p ∈ y.primeFactors, f p)
⊢ (ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) (x * y) =
    HMul.hMul (α := R) (β := R) ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) x : R)
      ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) y : R)
have rw₃ : ∏ p ∈ x.primeFactors ∪ y.primeFactors, f p = HMul.hMul (α := R) (∏ p ∈ x.primeFactors, f p) (∏ p ∈ y.primeFactors, f p)
Tactic: rw [prodPrimeFactors_apply hxy₀, prodPrimeFactors_apply hx, prodPrimeFactors_apply hy,
    primeFactors_mul hx hy, ← prod_union hxy.disjoint_primeFactors]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Misc (goal)
Goal:
R : Type u_1
inst : CommMonoidWithZero R
f : ℕ → R
x y : ℕ
hx : ¬x = 0
hy : ¬y = 0
hxy : x.Coprime y
hxy₀ : x * y ≠ 0
rw :
  ∏ p ∈ (x * y).primeFactors, f p =
    HMul.hMul (α := R) (β := R) ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) x : R)
      ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) y : R)
rw₁ :
  ∏ p ∈ (x * y).primeFactors, f p =
    HMul.hMul (α := R) (β := R) (∏ p ∈ x.primeFactors, f p : R)
      ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) y : R)
rw₂ : ∏ p ∈ (x * y).primeFactors, f p = HMul.hMul (α := R) (∏ p ∈ x.primeFactors, f p) (∏ p ∈ y.primeFactors, f p)
rw₃ :
  ∏ p ∈ x.primeFactors ∪ y.primeFactors, f p =
    HMul.hMul (α := R) (∏ p ∈ x.primeFactors, f p) (∏ p ∈ y.primeFactors, f p)
⊢ (ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) (x * y) =
    HMul.hMul (α := R) (β := R) ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) x : R)
      ((ArithmeticFunction.prodPrimeFactors f : (a : ℕ) → R) y : R)
have rw₄ : Eq.{u_1 + 1} (α := R) (∏ p ∈ x.primeFactors ∪ y.primeFactors, f p) (∏ x ∈ x.primeFactors ∪ y.primeFactors, f x)
Tactic: rw [prodPrimeFactors_apply hxy₀, prodPrimeFactors_apply hx, prodPrimeFactors_apply hy,
    primeFactors_mul hx hy, ← prod_union hxy.disjoint_primeFactors]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Misc (goal)
Goal:
R : Type u_1
inst : CommSemiring R
f g : ArithmeticFunction R
hf : IsMultiplicative.{u_1} (R := R) f
hg : IsMultiplicative.{u_1} (R := R) g
n : ℕ
hn : Squarefree n
⊢ Eq (α := R) ((ArithmeticFunction.prodPrimeFactors fun (p : ℕ) ↦ (f + g : (a : ℕ) → R) p : (a : ℕ) → R) n : R)
    ((f * g : (a : ℕ) → R) n : R)
have rw : Eq (α := R) (∏ p ∈ n.primeFactors, (f + g : (a : ℕ) → R) p : R) ((f * g : (a : ℕ) → R) n : R)
Tactic: rw [prodPrimeFactors_apply hn.ne_zero]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Misc (goal)
Goal:
R : Type u_1
inst : CommSemiring R
f g : ArithmeticFunction R
hf : IsMultiplicative.{u_1} (R := R) f
hg : IsMultiplicative.{u_1} (R := R) g
n : ℕ
hn : Squarefree n
⊢ Eq (α := R) (∏ p ∈ n.primeFactors, (f + g : (a : ℕ) → R) p : R) ((f * g : (a : ℕ) → R) n : R)
have simp_rw : Eq (α := R) (∏ x ∈ n.primeFactors, HAdd.hAdd (α := R) (β := R) ((f : (a : ℕ) → R) x : R) ((g : (a : ℕ) → R) x : R) : R)
  ((f * g : (a : ℕ) → R) n : R)
Tactic: simp_rw [add_apply (f := f) (g := g)]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Misc (goal)
Goal:
R : Type u_1
inst : CommSemiring R
f g : ArithmeticFunction R
hf : IsMultiplicative.{u_1} (R := R) f
hg : IsMultiplicative.{u_1} (R := R) g
n : ℕ
hn : Squarefree n
⊢ Eq (α := R)
    (∏ x ∈ n.primeFactors, HAdd.hAdd (α := R) (β := R) ((f : (a : ℕ) → R) x : R) ((g : (a : ℕ) → R) x : R) : R)
    ((f * g : (a : ℕ) → R) n : R)
have rw : Eq.{u_1 + 1} (α := R)
  (∑ t ∈ n.primeFactors.powerset,
    HMul.hMul (α := R) (∏ i ∈ t, (f : (a : ℕ) → R) i) (∏ i ∈ n.primeFactors \ t, (g : (a : ℕ) → R) i))
  (∑ i ∈ (↑n.primeFactorsList : Multiset ℕ).toFinset.powerset,
    HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) i.val.prod : R) ((g : (a : ℕ) → R) (n / i.val.prod) : R))
Tactic: rw [prod_add, mul_apply, sum_divisorsAntidiagonal (f · * g ·),
    ← divisors_filter_squarefree_of_squarefree hn, sum_divisors_filter_squarefree hn.ne_zero,
    factors_eq]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Misc (goal)
Goal:
R : Type u_1
inst : CommSemiring R
f g : ArithmeticFunction R
hf : IsMultiplicative.{u_1} (R := R) f
hg : IsMultiplicative.{u_1} (R := R) g
n : ℕ
hn : Squarefree n
⊢ Eq.{u_1 + 1} (α := R)
    (∑ t ∈ n.primeFactors.powerset,
      HMul.hMul (α := R) (∏ i ∈ t, (f : (a : ℕ) → R) i) (∏ i ∈ n.primeFactors \ t, (g : (a : ℕ) → R) i))
    (∑ i ∈ (↑n.primeFactorsList : Multiset ℕ).toFinset.powerset,
      HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) i.val.prod : R) ((g : (a : ℕ) → R) (n / i.val.prod) : R))
have apply : ∀ x ∈ n.primeFactors.powerset,
  HMul.hMul (α := R) (∏ i ∈ x, (f : (a : ℕ) → R) i) (∏ i ∈ n.primeFactors \ x, (g : (a : ℕ) → R) i) =
    HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) x.val.prod : R) ((g : (a : ℕ) → R) (n / x.val.prod) : R)
Tactic: apply sum_congr rfl
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Misc (goal)
Goal:
R : Type u_1
inst : CommSemiring R
f g : ArithmeticFunction R
hf : IsMultiplicative.{u_1} (R := R) f
hg : IsMultiplicative.{u_1} (R := R) g
n : ℕ
hn : Squarefree n
t : Finset ℕ
ht : t ∈ n.primeFactors.powerset
⊢ HMul.hMul (α := R) (∏ i ∈ t, (f : (a : ℕ) → R) i) (∏ i ∈ n.primeFactors \ t, (g : (a : ℕ) → R) i) =
    HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) t.val.prod : R) ((g : (a : ℕ) → R) (n / t.val.prod) : R)
have rw : HMul.hMul (α := R) (∏ i ∈ t, (f : (a : ℕ) → R) i) (∏ i ∈ n.primeFactors \ t, (g : (a : ℕ) → R) i) =
  HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) (t.prod id) : R) ((g : (a : ℕ) → R) (n / t.prod id) : R)
Tactic: rw [t.prod_val, Function.id_def,
    ← prod_primeFactors_sdiff_of_squarefree hn (mem_powerset.mp ht),
    hf.map_prod_of_subset_primeFactors n t (mem_powerset.mp ht),
    ← hg.map_prod_of_subset_primeFactors n (_ \ t) sdiff_subset]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Misc (goal)
Goal:
R : Type u_1
inst : CommSemiring R
f g : ArithmeticFunction R
hf : IsMultiplicative.{u_1} (R := R) f
hg : IsMultiplicative.{u_1} (R := R) g
n : ℕ
hn : Squarefree n
t : Finset ℕ
ht : t ∈ n.primeFactors.powerset
rw :
  HMul.hMul (α := R) (∏ i ∈ t, (f : (a : ℕ) → R) i) (∏ i ∈ n.primeFactors \ t, (g : (a : ℕ) → R) i) =
    HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) (t.prod id) : R) ((g : (a : ℕ) → R) (n / t.prod id) : R)
⊢ HMul.hMul (α := R) (∏ i ∈ t, (f : (a : ℕ) → R) i) (∏ i ∈ n.primeFactors \ t, (g : (a : ℕ) → R) i) =
    HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) t.val.prod : R) ((g : (a : ℕ) → R) (n / t.val.prod) : R)
have rw₁ : HMul.hMul (α := R) (∏ i ∈ t, (f : (a : ℕ) → R) i) (∏ i ∈ n.primeFactors \ t, (g : (a : ℕ) → R) i) =
  HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) (∏ x ∈ t, x) : R) ((g : (a : ℕ) → R) (n / ∏ x ∈ t, x) : R)
Tactic: rw [t.prod_val, Function.id_def,
    ← prod_primeFactors_sdiff_of_squarefree hn (mem_powerset.mp ht),
    hf.map_prod_of_subset_primeFactors n t (mem_powerset.mp ht),
    ← hg.map_prod_of_subset_primeFactors n (_ \ t) sdiff_subset]