awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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
this_1 : UniformSpace R := ⊥
this : IsUniformInducing.{u_2, u_2} (α := ArithmeticFunction R) (β := (a : ℕ) → R) (DFunLike.coe (β := fun (x : ℕ) ↦ R))
n : ℕ
s : Set ι
hs : Membership.mem (γ := Filter ι) cofinite s
hs' : ∀ y ∈ s, ∀ i ∈ Set.Iic n, Eq (α := R) ((f y : (a : ℕ) → R) i : R) ((1 : (a : ℕ) → R) i : R)
t : Finset ι := ⋯.toFinset (s := sᶜ)
u v : Finset ι
hu : ∀ i ∈ u \ t, i ∈ s
hv : ∀ i ∈ v \ t, i ∈ s
i : ι
w : Finset ι
hi : i ∉ w
hw' :
  ∀ (hw : ∀ i ∈ w, i ∈ s), ∀ k ≤ n, Eq.{u_2 + 1} (α := R) ((∏ x ∈ w, f x : (a : ℕ) → R) k : R) ((1 : (a : ℕ) → R) k : R)
hw : ∀ i_1 ∈ insert i w, i_1 ∈ s
k : ℕ
hk : k ≤ n
⊢ Eq.{u_2 + 1} (α := R)
    (∑ x ∈ k.divisorsAntidiagonal,
      HMul.hMul (α := R) (β := R) ((f i : (a : ℕ) → R) x.1 : R) ((∏ x ∈ w, f x : (a : ℕ) → R) x.2 : R))
    (∑ x ∈ k.divisorsAntidiagonal, HMul.hMul (α := R) (β := R) ((1 : (a : ℕ) → R) x.1 : R) ((1 : (a : ℕ) → R) x.2 : R))
have refine : ∀ (j : ℕ × ℕ) (hj : j ∈ k.divisorsAntidiagonal),
  HMul.hMul (α := R) (β := R) ((f i : (a : ℕ) → R) j.1 : R) ((∏ x ∈ w, f x : (a : ℕ) → R) j.2 : R) =
    HMul.hMul (α := R) (β := R) ((1 : (a : ℕ) → R) j.1 : R) ((1 : (a : ℕ) → R) j.2 : R)
Tactic: refine Finset.sum_congr rfl fun j hj ↦ ?_
-- 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
this_1 : UniformSpace R := ⊥
this : IsUniformInducing.{u_2, u_2} (α := ArithmeticFunction R) (β := (a : ℕ) → R) (DFunLike.coe (β := fun (x : ℕ) ↦ R))
n : ℕ
s : Set ι
hs : Membership.mem (γ := Filter ι) cofinite s
hs' : ∀ y ∈ s, ∀ i ∈ Set.Iic n, Eq (α := R) ((f y : (a : ℕ) → R) i : R) ((1 : (a : ℕ) → R) i : R)
t : Finset ι := ⋯.toFinset (s := sᶜ)
u v : Finset ι
hu : ∀ i ∈ u \ t, i ∈ s
hv : ∀ i ∈ v \ t, i ∈ s
i : ι
w : Finset ι
hi : i ∉ w
hw' :
  ∀ (hw : ∀ i ∈ w, i ∈ s), ∀ k ≤ n, Eq.{u_2 + 1} (α := R) ((∏ x ∈ w, f x : (a : ℕ) → R) k : R) ((1 : (a : ℕ) → R) k : R)
hw : ∀ i_1 ∈ insert i w, i_1 ∈ s
k : ℕ
hk : k ≤ n
j : ℕ × ℕ
hj : j ∈ k.divisorsAntidiagonal
⊢ HMul.hMul (α := R) (β := R) ((f i : (a : ℕ) → R) j.1 : R) ((∏ x ∈ w, f x : (a : ℕ) → R) j.2 : R) =
    HMul.hMul (α := R) (β := R) ((1 : (a : ℕ) → R) j.1 : R) ((1 : (a : ℕ) → R) j.2 : R)
have h1 : Eq (α := R) ((f i : (a : ℕ) → R) j.1 : R) ((1 : (a : ℕ) → R) j.1 : R)
Tactic: have h1 := hs' i (hw i (Finset.mem_insert_self i w)) j.1
      ((Nat.divisor_le (Nat.fst_mem_divisors_of_mem_antidiagonal hj)).trans hk)
-- 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
this_1 : UniformSpace R := ⊥
this : IsUniformInducing.{u_2, u_2} (α := ArithmeticFunction R) (β := (a : ℕ) → R) (DFunLike.coe (β := fun (x : ℕ) ↦ R))
n : ℕ
s : Set ι
hs : Membership.mem (γ := Filter ι) cofinite s
hs' : ∀ y ∈ s, ∀ i ∈ Set.Iic n, Eq (α := R) ((f y : (a : ℕ) → R) i : R) ((1 : (a : ℕ) → R) i : R)
t : Finset ι := ⋯.toFinset (s := sᶜ)
u v : Finset ι
hu : ∀ i ∈ u \ t, i ∈ s
hv : ∀ i ∈ v \ t, i ∈ s
i : ι
w : Finset ι
hi : i ∉ w
hw' :
  ∀ (hw : ∀ i ∈ w, i ∈ s), ∀ k ≤ n, Eq.{u_2 + 1} (α := R) ((∏ x ∈ w, f x : (a : ℕ) → R) k : R) ((1 : (a : ℕ) → R) k : R)
hw : ∀ i_1 ∈ insert i w, i_1 ∈ s
k : ℕ
hk : k ≤ n
j : ℕ × ℕ
hj : j ∈ k.divisorsAntidiagonal
h1 : Eq (α := R) ((f i : (a : ℕ) → R) j.1 : R) ((1 : (a : ℕ) → R) j.1 : R)
⊢ HMul.hMul (α := R) (β := R) ((f i : (a : ℕ) → R) j.1 : R) ((∏ x ∈ w, f x : (a : ℕ) → R) j.2 : R) =
    HMul.hMul (α := R) (β := R) ((1 : (a : ℕ) → R) j.1 : R) ((1 : (a : ℕ) → R) j.2 : R)
have h2 : Eq.{u_2 + 1} (α := R) ((∏ x ∈ w, f x : (a : ℕ) → R) j.2 : R) ((1 : (a : ℕ) → R) j.2 : R)
Tactic: have h2 := hw' (fun i hi ↦ hw i (Finset.mem_insert_of_mem hi)) j.2
      ((Nat.divisor_le (Nat.snd_mem_divisors_of_mem_antidiagonal hj)).trans hk)
-- 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
this_1 : UniformSpace R := ⊥
this : IsUniformInducing.{u_2, u_2} (α := ArithmeticFunction R) (β := (a : ℕ) → R) (DFunLike.coe (β := fun (x : ℕ) ↦ R))
n : ℕ
s : Set ι
hs : Membership.mem (γ := Filter ι) cofinite s
hs' : ∀ y ∈ s, ∀ i ∈ Set.Iic n, Eq (α := R) ((f y : (a : ℕ) → R) i : R) ((1 : (a : ℕ) → R) i : R)
t : Finset ι := ⋯.toFinset (s := sᶜ)
u v : Finset ι
hu : ∀ i ∈ u \ t, i ∈ s
hv : ∀ i ∈ v \ t, i ∈ s
i : ι
w : Finset ι
hi : i ∉ w
hw' :
  ∀ (hw : ∀ i ∈ w, i ∈ s), ∀ k ≤ n, Eq.{u_2 + 1} (α := R) ((∏ x ∈ w, f x : (a : ℕ) → R) k : R) ((1 : (a : ℕ) → R) k : R)
hw : ∀ i_1 ∈ insert i w, i_1 ∈ s
k : ℕ
hk : k ≤ n
j : ℕ × ℕ
hj : j ∈ k.divisorsAntidiagonal
h1 : Eq (α := R) ((f i : (a : ℕ) → R) j.1 : R) ((1 : (a : ℕ) → R) j.1 : R)
h2 : Eq.{u_2 + 1} (α := R) ((∏ x ∈ w, f x : (a : ℕ) → R) j.2 : R) ((1 : (a : ℕ) → R) j.2 : R)
⊢ HMul.hMul (α := R) (β := R) ((f i : (a : ℕ) → R) j.1 : R) ((∏ x ∈ w, f x : (a : ℕ) → R) j.2 : R) =
    HMul.hMul (α := R) (β := R) ((1 : (a : ℕ) → R) j.1 : R) ((1 : (a : ℕ) → R) j.2 : R)
have rw : HMul.hMul (α := R) (β := R) ((1 : (a : ℕ) → R) j.1 : R) ((∏ x ∈ w, f x : (a : ℕ) → R) j.2 : R) =
  HMul.hMul (α := R) (β := R) ((1 : (a : ℕ) → R) j.1 : R) ((1 : (a : ℕ) → R) j.2 : R)
Tactic: rw [h1, 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
this_1 : UniformSpace R := ⊥
this : IsUniformInducing.{u_2, u_2} (α := ArithmeticFunction R) (β := (a : ℕ) → R) (DFunLike.coe (β := fun (x : ℕ) ↦ R))
n : ℕ
s : Set ι
hs : Membership.mem (γ := Filter ι) cofinite s
hs' : ∀ y ∈ s, ∀ i ∈ Set.Iic n, Eq (α := R) ((f y : (a : ℕ) → R) i : R) ((1 : (a : ℕ) → R) i : R)
t : Finset ι := ⋯.toFinset (s := sᶜ)
u v : Finset ι
hu : ∀ i ∈ u \ t, i ∈ s
hv : ∀ i ∈ v \ t, i ∈ s
i : ι
w : Finset ι
hi : i ∉ w
hw' :
  ∀ (hw : ∀ i ∈ w, i ∈ s), ∀ k ≤ n, Eq.{u_2 + 1} (α := R) ((∏ x ∈ w, f x : (a : ℕ) → R) k : R) ((1 : (a : ℕ) → R) k : R)
hw : ∀ i_1 ∈ insert i w, i_1 ∈ s
k : ℕ
hk : k ≤ n
j : ℕ × ℕ
hj : j ∈ k.divisorsAntidiagonal
h1 : Eq (α := R) ((f i : (a : ℕ) → R) j.1 : R) ((1 : (a : ℕ) → R) j.1 : R)
h2 : Eq.{u_2 + 1} (α := R) ((∏ x ∈ w, f x : (a : ℕ) → R) j.2 : R) ((1 : (a : ℕ) → R) j.2 : R)
rw :
  HMul.hMul (α := R) (β := R) ((1 : (a : ℕ) → R) j.1 : R) ((∏ x ∈ w, f x : (a : ℕ) → R) j.2 : R) =
    HMul.hMul (α := R) (β := R) ((1 : (a : ℕ) → R) j.1 : R) ((1 : (a : ℕ) → R) j.2 : R)
⊢ HMul.hMul (α := R) (β := R) ((f i : (a : ℕ) → R) j.1 : R) ((∏ x ∈ w, f x : (a : ℕ) → R) j.2 : R) =
    HMul.hMul (α := R) (β := R) ((1 : (a : ℕ) → R) j.1 : R) ((1 : (a : ℕ) → R) j.2 : R)
have rw₁ : HMul.hMul (α := R) (β := R) ((1 : (a : ℕ) → R) j.1 : R) ((1 : (a : ℕ) → R) j.2 : R) =
  HMul.hMul (α := R) (β := R) ((1 : (a : ℕ) → R) j.1 : R) ((1 : (a : ℕ) → R) j.2 : R)
Tactic: rw [h1, 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 : ∀ (n : ℕ), ∀ᶠ (i : ι) in cofinite, Eq (α := R) ((f i : (a : ℕ) → R) n : R) ((1 : (a : ℕ) → R) n : R)
this_1 : UniformSpace R := ⊥
this : IsUniformInducing.{u_2, u_2} (α := ArithmeticFunction R) (β := (a : ℕ) → R) (DFunLike.coe (β := fun (x : ℕ) ↦ R))
⊢ ∀ (n : ℕ),
    ∀ᶠ (s : Finset ι) in atTop, Eq (α := R) ((∏ i ∈ s, f i : (a : ℕ) → R) n : R) ((eulerProduct f : (a : ℕ) → R) n : R)
have suffices : Multipliable f
Tactic: classical
  suffices Multipliable f from tendsto_iff.mp this.hasProd
  simp_rw [multipliable_iff_cauchySeq_finset, CauchySeq, ← this.cauchy_map_iff,
    Filter.map_map, cauchy_map_iff', Pi.uniformity, DiscreteUniformity.eq_principal_setRelId,
    tendsto_iInf, tendsto_comap_iff, tendsto_principal, Function.comp_apply, prod_atTop_atTop_eq,
    eventually_atTop_prod_self, SetRel.mem_id]
  intro n
  replace hf : ∀ k ∈ Set.Iic n, ∀ᶠ (x : ι) in cofinite, (f x) k = (1 : ArithmeticFunction R) k :=
    fun k hk ↦ hf k
  rw [← eventually_all_finite (Set.finite_Iic n), eventually_iff_exists_mem] at hf
  obtain ⟨s, hs, hs'⟩ := hf
  let t := (mem_cofinite.mp hs).toFinset
  refine ⟨t, fun u v hu hv ↦ ?_⟩
  rw [← Finset.prod_sdiff hu, ← Finset.prod_sdiff hv]
  replace hu : ∀ i ∈ u \ t, i ∈ s := by
    intro i hi
    rw [Finset.mem_sdiff, Set.Finite.mem_toFinset, Set.notMem_compl_iff] at hi
    exact hi.2
  replace hv : ∀ i ∈ v \ t, i ∈ s := by
    intro i hi
    rw [Finset.mem_sdiff, Set.Finite.mem_toFinset, Set.notMem_compl_iff] at hi
    exact hi.2
  suffices ∀ k ≤ n, (∏ x ∈ u \ t, f x) k = (∏ x ∈ v \ t, f x) k by
    rw [mul_apply, mul_apply]
    refine Finset.sum_congr rfl fun k hk ↦ ?_
    rw [this k.1 (Nat.divisor_le (Nat.fst_mem_divisors_of_mem_antidiagonal hk))]
  suffices ∀ w, (∀ i ∈ w, i ∈ s) → ∀ k ≤ n, (∏ x ∈ w, f x) k = (1 : ArithmeticFunction R) k by
    intro k hk
    rw [this (u \ t) hu k hk, this (v \ t) hv k hk]
  intro w hw
  induction w using Finset.induction_on
  case empty => simp
  case insert i w hi hw' =>
    intro k hk
    rw [← one_mul (1 : ArithmeticFunction R), Finset.prod_insert hi, mul_apply, mul_apply]
    refine Finset.sum_congr rfl fun j hj ↦ ?_
    have h1 := hs' i (hw i (Finset.mem_insert_self i w)) j.1
      ((Nat.divisor_le (Nat.fst_mem_divisors_of_mem_antidiagonal hj)).trans hk)
    have h2 := hw' (fun i hi ↦ hw i (Finset.mem_insert_of_mem hi)) j.2
      ((Nat.divisor_le (Nat.snd_mem_divisors_of_mem_antidiagonal hj)).trans hk)
    rw [h1, 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 : ∀ (n : ℕ), ∀ᶠ (i : ι) in cofinite, Eq (α := R) ((f i : (a : ℕ) → R) n : R) ((1 : (a : ℕ) → R) n : R)
this_1 : UniformSpace R := ⊥
this : IsUniformInducing.{u_2, u_2} (α := ArithmeticFunction R) (β := (a : ℕ) → R) (DFunLike.coe (β := fun (x : ℕ) ↦ R))
suffices suffices₁ suffices₂ : Multipliable f
⊢ ∀ (n : ℕ),
    ∀ᶠ (s : Finset ι) in atTop, Eq (α := R) ((∏ i ∈ s, f i : (a : ℕ) → R) n : R) ((eulerProduct f : (a : ℕ) → R) n : R)
have suffices₃ : ∀ (i : ℕ),
  ∃ (a : Finset ι),
    ∀ (k l : Finset ι),
      a ≤ k → a ≤ l → Eq.{u_2 + 1} (α := R) ((∏ b ∈ k, f b : (a : ℕ) → R) i : R) ((∏ b ∈ l, f b : (a : ℕ) → R) i : R)
Tactic: classical
  suffices Multipliable f from tendsto_iff.mp this.hasProd
  simp_rw [multipliable_iff_cauchySeq_finset, CauchySeq, ← this.cauchy_map_iff,
    Filter.map_map, cauchy_map_iff', Pi.uniformity, DiscreteUniformity.eq_principal_setRelId,
    tendsto_iInf, tendsto_comap_iff, tendsto_principal, Function.comp_apply, prod_atTop_atTop_eq,
    eventually_atTop_prod_self, SetRel.mem_id]
  intro n
  replace hf : ∀ k ∈ Set.Iic n, ∀ᶠ (x : ι) in cofinite, (f x) k = (1 : ArithmeticFunction R) k :=
    fun k hk ↦ hf k
  rw [← eventually_all_finite (Set.finite_Iic n), eventually_iff_exists_mem] at hf
  obtain ⟨s, hs, hs'⟩ := hf
  let t := (mem_cofinite.mp hs).toFinset
  refine ⟨t, fun u v hu hv ↦ ?_⟩
  rw [← Finset.prod_sdiff hu, ← Finset.prod_sdiff hv]
  replace hu : ∀ i ∈ u \ t, i ∈ s := by
    intro i hi
    rw [Finset.mem_sdiff, Set.Finite.mem_toFinset, Set.notMem_compl_iff] at hi
    exact hi.2
  replace hv : ∀ i ∈ v \ t, i ∈ s := by
    intro i hi
    rw [Finset.mem_sdiff, Set.Finite.mem_toFinset, Set.notMem_compl_iff] at hi
    exact hi.2
  suffices ∀ k ≤ n, (∏ x ∈ u \ t, f x) k = (∏ x ∈ v \ t, f x) k by
    rw [mul_apply, mul_apply]
    refine Finset.sum_congr rfl fun k hk ↦ ?_
    rw [this k.1 (Nat.divisor_le (Nat.fst_mem_divisors_of_mem_antidiagonal hk))]
  suffices ∀ w, (∀ i ∈ w, i ∈ s) → ∀ k ≤ n, (∏ x ∈ w, f x) k = (1 : ArithmeticFunction R) k by
    intro k hk
    rw [this (u \ t) hu k hk, this (v \ t) hv k hk]
  intro w hw
  induction w using Finset.induction_on
  case empty => simp
  case insert i w hi hw' =>
    intro k hk
    rw [← one_mul (1 : ArithmeticFunction R), Finset.prod_insert hi, mul_apply, mul_apply]
    refine Finset.sum_congr rfl fun j hj ↦ ?_
    have h1 := hs' i (hw i (Finset.mem_insert_self i w)) j.1
      ((Nat.divisor_le (Nat.fst_mem_divisors_of_mem_antidiagonal hj)).trans hk)
    have h2 := hw' (fun i hi ↦ hw i (Finset.mem_insert_of_mem hi)) j.2
      ((Nat.divisor_le (Nat.snd_mem_divisors_of_mem_antidiagonal hj)).trans hk)
    rw [h1, 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 : ∀ (n : ℕ), ∀ᶠ (i : ι) in cofinite, Eq (α := R) ((f i : (a : ℕ) → R) n : R) ((1 : (a : ℕ) → R) n : R)
this_1 : UniformSpace R := ⊥
this : IsUniformInducing.{u_2, u_2} (α := ArithmeticFunction R) (β := (a : ℕ) → R) (DFunLike.coe (β := fun (x : ℕ) ↦ R))
suffices suffices₁ suffices₂ : Multipliable f
suffices₃ suffices₄ :
  ∀ (n : ℕ),
    ∃ (a : Finset ι),
      ∀ (k l : Finset ι),
        a ≤ k → a ≤ l → Eq.{u_2 + 1} (α := R) ((∏ b ∈ k, f b : (a : ℕ) → R) n : R) ((∏ b ∈ l, f b : (a : ℕ) → R) n : R)
⊢ ∀ (n : ℕ),
    ∀ᶠ (s : Finset ι) in atTop, Eq (α := R) ((∏ i ∈ s, f i : (a : ℕ) → R) n : R) ((eulerProduct f : (a : ℕ) → R) n : R)
have suffices₅ : ∀ (n : ℕ)
  (hf : ∀ k ∈ Set.Iic n, ∀ᶠ (x : ι) in cofinite, Eq (α := R) ((f x : (a : ℕ) → R) k : R) ((1 : (a : ℕ) → R) k : R)),
  ∃ (a : Finset ι),
    ∀ (k l : Finset ι),
      a ≤ k → a ≤ l → Eq.{u_2 + 1} (α := R) ((∏ b ∈ k, f b : (a : ℕ) → R) n : R) ((∏ b ∈ l, f b : (a : ℕ) → R) n : R)
Tactic: classical
  suffices Multipliable f from tendsto_iff.mp this.hasProd
  simp_rw [multipliable_iff_cauchySeq_finset, CauchySeq, ← this.cauchy_map_iff,
    Filter.map_map, cauchy_map_iff', Pi.uniformity, DiscreteUniformity.eq_principal_setRelId,
    tendsto_iInf, tendsto_comap_iff, tendsto_principal, Function.comp_apply, prod_atTop_atTop_eq,
    eventually_atTop_prod_self, SetRel.mem_id]
  intro n
  replace hf : ∀ k ∈ Set.Iic n, ∀ᶠ (x : ι) in cofinite, (f x) k = (1 : ArithmeticFunction R) k :=
    fun k hk ↦ hf k
  rw [← eventually_all_finite (Set.finite_Iic n), eventually_iff_exists_mem] at hf
  obtain ⟨s, hs, hs'⟩ := hf
  let t := (mem_cofinite.mp hs).toFinset
  refine ⟨t, fun u v hu hv ↦ ?_⟩
  rw [← Finset.prod_sdiff hu, ← Finset.prod_sdiff hv]
  replace hu : ∀ i ∈ u \ t, i ∈ s := by
    intro i hi
    rw [Finset.mem_sdiff, Set.Finite.mem_toFinset, Set.notMem_compl_iff] at hi
    exact hi.2
  replace hv : ∀ i ∈ v \ t, i ∈ s := by
    intro i hi
    rw [Finset.mem_sdiff, Set.Finite.mem_toFinset, Set.notMem_compl_iff] at hi
    exact hi.2
  suffices ∀ k ≤ n, (∏ x ∈ u \ t, f x) k = (∏ x ∈ v \ t, f x) k by
    rw [mul_apply, mul_apply]
    refine Finset.sum_congr rfl fun k hk ↦ ?_
    rw [this k.1 (Nat.divisor_le (Nat.fst_mem_divisors_of_mem_antidiagonal hk))]
  suffices ∀ w, (∀ i ∈ w, i ∈ s) → ∀ k ≤ n, (∏ x ∈ w, f x) k = (1 : ArithmeticFunction R) k by
    intro k hk
    rw [this (u \ t) hu k hk, this (v \ t) hv k hk]
  intro w hw
  induction w using Finset.induction_on
  case empty => simp
  case insert i w hi hw' =>
    intro k hk
    rw [← one_mul (1 : ArithmeticFunction R), Finset.prod_insert hi, mul_apply, mul_apply]
    refine Finset.sum_congr rfl fun j hj ↦ ?_
    have h1 := hs' i (hw i (Finset.mem_insert_self i w)) j.1
      ((Nat.divisor_le (Nat.fst_mem_divisors_of_mem_antidiagonal hj)).trans hk)
    have h2 := hw' (fun i hi ↦ hw i (Finset.mem_insert_of_mem hi)) j.2
      ((Nat.divisor_le (Nat.snd_mem_divisors_of_mem_antidiagonal hj)).trans hk)
    rw [h1, h2]
Cleared hypotheses: hf
-- 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)
⊢ IsMultiplicative.{u_2} (R := R) (eulerProduct f)
have pos : ∀ (hf' : Multipliable f), IsMultiplicative.{u_2} (R := R) (eulerProduct f)
Tactic: by_cases hf' : Multipliable f
-- 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)
pos : ∀ (hf' : Multipliable f), IsMultiplicative.{u_2} (R := R) (eulerProduct f)
⊢ IsMultiplicative.{u_2} (R := R) (eulerProduct f)
have neg : ∀ (hf' : ¬Multipliable f), IsMultiplicative.{u_2} (R := R) (eulerProduct f)
Tactic: by_cases hf' : Multipliable f
-- 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
⊢ IsMultiplicative.{u_2} (R := R) (eulerProduct f)
have h : ∀ (s : Finset ι), IsMultiplicative.{u_2} (R := R) (∏ b ∈ s, f b)
Tactic: have h (s : Finset ι) : (∏ b ∈ s, f b).IsMultiplicative :=
      isMultiplicative_finsetProd f s fun i a ↦ hf i
-- 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 : ∀ (s : Finset ι), IsMultiplicative.{u_2} (R := R) (∏ b ∈ s, f b)
⊢ IsMultiplicative.{u_2} (R := R) (eulerProduct f)
have 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)
Tactic: have key := tendsto_iff.mp hf'.hasProd
-- 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
h : ∀ (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)
⊢ IsMultiplicative.{u_2} (R := R) (eulerProduct f)
have pos : ∀ (h : ∀ (x : Finset ι), Eq.{u_2 + 1} (α := R) ((∏ b ∈ x, f b : (a : ℕ) → R) 1 : R) (1 : R)),
  Eq (α := R) ((eulerProduct f : (a : ℕ) → R) 1 : R) (1 : R)
Tactic: refine (forall_and.mp h).imp (fun h ↦ ?_) fun h m n hmn ↦ ?_
-- 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
h : ∀ (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)
pos :
  ∀ (h : ∀ (x : Finset ι), Eq.{u_2 + 1} (α := R) ((∏ b ∈ x, f b : (a : ℕ) → R) 1 : R) (1 : R)),
    Eq (α := R) ((eulerProduct f : (a : ℕ) → R) 1 : R) (1 : R)
⊢ IsMultiplicative.{u_2} (R := R) (eulerProduct f)
have pos₁ : ∀
  (h :
    ∀ (x : Finset ι) {m n : ℕ},
      m.Coprime n →
        (∏ b ∈ x, f b : (a : ℕ) → R) (m * n) =
          HMul.hMul (α := R) (β := R) ((∏ b ∈ x, f b : (a : ℕ) → R) m : R) ((∏ b ∈ x, f b : (a : ℕ) → R) n : R))
  (m n : ℕ) (hmn : m.Coprime n),
  (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)
Tactic: refine (forall_and.mp h).imp (fun h ↦ ?_) fun h m n hmn ↦ ?_
-- 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)
h : ∀ (x : Finset ι), Eq.{u_2 + 1} (α := R) ((∏ b ∈ x, f b : (a : ℕ) → R) 1 : R) (1 : R)
⊢ Eq (α := R) ((eulerProduct f : (a : ℕ) → R) 1 : R) (1 : R)
have key₁ : ∀ᶠ (i : Finset ι) in (SummationFilter.unconditional ι).filter,
  Eq.{u_2 + 1} (α := R) ((∏ b ∈ i, f b : (a : ℕ) → R) 1 : R) ((∏' (b : ι), f b : (a : ℕ) → R) 1 : R)
Tactic: specialize key 1
Cleared hypotheses: key
-- 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)
h : ∀ (x : Finset ι), Eq.{u_2 + 1} (α := R) ((∏ b ∈ x, f b : (a : ℕ) → R) 1 : R) (1 : R)
key :
  ∀ᶠ (i : Finset ι) in (SummationFilter.unconditional ι).filter,
    Eq.{u_2 + 1} (α := R) ((∏ b ∈ i, f b : (a : ℕ) → R) 1 : R) ((∏' (b : ι), f b : (a : ℕ) → R) 1 : R)
⊢ Eq (α := R) ((eulerProduct f : (a : ℕ) → R) 1 : R) (1 : R)
have key₁ : ∀ᶠ (i : Finset ι) in (SummationFilter.unconditional ι).filter,
  Eq.{u_2 + 1} (α := R) (1 : R) ((∏' (b : ι), f b : (a : ℕ) → R) 1 : R)
Tactic: simp_rw [h] at key
Cleared hypotheses: key
-- 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
h_1 : ∀ (s : Finset ι), IsMultiplicative.{u_2} (R := R) (∏ b ∈ s, f b)
h : ∀ (x : Finset ι), Eq.{u_2 + 1} (α := R) ((∏ b ∈ x, f b : (a : ℕ) → R) 1 : R) (1 : R)
key :
  ∀ᶠ (i : Finset ι) in (SummationFilter.unconditional ι).filter,
    Eq.{u_2 + 1} (α := R) (1 : R) ((∏' (b : ι), f b : (a : ℕ) → R) 1 : R)
⊢ Eq (α := R) ((eulerProduct f : (a : ℕ) → R) 1 : R) (1 : R)
have pos : ∀ (key : Eq.{u_2 + 1} (α := R) (1 : R) ((∏' (b : ι), f b : (a : ℕ) → R) 1 : R)),
  Eq (α := R) ((eulerProduct f : (a : ℕ) → R) 1 : R) (1 : R)
Tactic: rwa [eventually_const, eq_comm] at key
Cleared hypotheses: key
-- 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
h_1 : ∀ (s : Finset ι), IsMultiplicative.{u_2} (R := R) (∏ b ∈ s, f b)
h : ∀ (x : Finset ι), Eq.{u_2 + 1} (α := R) ((∏ b ∈ x, f b : (a : ℕ) → R) 1 : R) (1 : R)
key :
  ∀ᶠ (i : Finset ι) in (SummationFilter.unconditional ι).filter,
    Eq.{u_2 + 1} (α := R) (1 : R) ((∏' (b : ι), f b : (a : ℕ) → R) 1 : R)
pos :
  ∀ (key : Eq.{u_2 + 1} (α := R) (1 : R) ((∏' (b : ι), f b : (a : ℕ) → R) 1 : R)),
    Eq (α := R) ((eulerProduct f : (a : ℕ) → R) 1 : R) (1 : R)
⊢ Eq (α := R) ((eulerProduct f : (a : ℕ) → R) 1 : R) (1 : R)
have pos₁ : ∀ (key : Eq.{u_2 + 1} (α := R) ((∏' (b : ι), f b : (a : ℕ) → R) 1 : R) (1 : R)),
  Eq (α := R) ((eulerProduct f : (a : ℕ) → R) 1 : R) (1 : R)
Tactic: rwa [eventually_const, eq_comm] at key
Cleared hypotheses: key
-- 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)
h :
  ∀ (x : Finset ι) {m n : ℕ},
    m.Coprime n →
      (∏ b ∈ x, f b : (a : ℕ) → R) (m * n) =
        HMul.hMul (α := R) (β := R) ((∏ b ∈ x, f b : (a : ℕ) → R) m : R) ((∏ b ∈ x, f b : (a : ℕ) → R) n : R)
m n : ℕ
hmn : m.Coprime n
⊢ (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 h₁ : ∀ (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)
Tactic: replace h s : (∏ b ∈ s, f b) (m * n) = (∏ b ∈ s, f b) m * (∏ b ∈ s, f b) n := h s hmn
Cleared hypotheses: h