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
this : UniformSpace R := ⊥
⊢ IsComplete.{u_2} (α := (a : ℕ) → R) (Set.range (ι := ArithmeticFunction R) (DFunLike.coe (β := fun (x : ℕ) ↦ R)))
have h : IsClosed.{u_2} (X := (a : ℕ) → R) (Set.range (ι := ArithmeticFunction R) (DFunLike.coe (β := fun (x : ℕ) ↦ R)))
Tactic: apply IsClosed.isComplete
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (goal)
Goal:
ι : Type u_1
R : Type u_2
inst : CommSemiring R
this : UniformSpace R := ⊥
⊢ Eq.{u_2 + 1} (α := Set ((a : ℕ) → R)) (Set.range (ι := ArithmeticFunction R) (DFunLike.coe (β := fun (x : ℕ) ↦ R)))
    {f : (a : ℕ) → R | f 0 = 0}
have h : ∀ (f : (a : ℕ) → R),
  Membership.mem (γ := Set ((a : ℕ) → R)) (Set.range (ι := ArithmeticFunction R) (DFunLike.coe (β := fun (x : ℕ) ↦ R)))
      f ↔
    Membership.mem (γ := Set ((a : ℕ) → R)) {f : (a : ℕ) → R | f 0 = 0} f
Tactic: ext f
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (hypothesis)
Goal:
ι : Type u_1
R : Type u_2
inst : CommSemiring R
this : UniformSpace R := ⊥
⊢ IsClosed.{u_2} (X := (a : ℕ) → R) (Set.range (ι := ArithmeticFunction R) (DFunLike.coe (β := fun (x : ℕ) ↦ R)))
have this₁ : Eq.{u_2 + 1} (α := Set ((a : ℕ) → R)) (Set.range (ι := ArithmeticFunction R) (DFunLike.coe (β := fun (x : ℕ) ↦ R)))
  {f : (a : ℕ) → R | f 0 = 0}
Tactic: have : Set.range ((↑) : ArithmeticFunction R → (ℕ → R)) = {f | f 0 = 0} := by
    ext f
    exact ⟨by rintro ⟨f, rfl⟩; simp, fun hf ↦ ⟨⟨f, hf⟩, rfl⟩⟩
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.LFunction (goal)
Goal:
ι : Type u_1
R : Type u_2
inst : CommSemiring R
this_1 : UniformSpace R := ⊥
this :
  Eq.{u_2 + 1} (α := Set ((a : ℕ) → R)) (Set.range (ι := ArithmeticFunction R) (DFunLike.coe (β := fun (x : ℕ) ↦ R)))
    {f : (a : ℕ) → R | f 0 = 0}
⊢ IsClosed.{u_2} (X := (a : ℕ) → R) (Set.range (ι := ArithmeticFunction R) (DFunLike.coe (β := fun (x : ℕ) ↦ R)))
have h : IsClosed.{u_2} (X := (a : ℕ) → R) {f : (a : ℕ) → R | f 0 = 0}
Tactic: rw [ArithmeticFunction.range_coe]
-- 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 : ∀ (n : ℕ), ∀ᶠ (i : ι) in cofinite, Eq (α := R) ((f i : (a : ℕ) → R) n : R) ((1 : (a : ℕ) → R) n : R)
this : UniformSpace 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 this₁ : IsUniformInducing.{u_2, u_2} (α := ArithmeticFunction R) (β := (a : ℕ) → R) (DFunLike.coe (β := fun (x : ℕ) ↦ R))
Tactic: have : IsUniformInducing ((↑) : ArithmeticFunction R → (ℕ → R)) := ⟨rfl⟩
-- 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: suffices Multipliable f from tendsto_iff.mp this.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 : ∀ (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))
⊢ Multipliable f
have simp_rw : ∀ (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: 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]
-- 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 : ∀ (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
⊢ ∃ (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)
have hf₁ : ∀ 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)
Tactic: replace hf : ∀ k ∈ Set.Iic n, ∀ᶠ (x : ι) in cofinite, (f x) k = (1 : ArithmeticFunction R) k :=
    fun k hk ↦ hf k
Cleared hypotheses: hf
-- 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
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)
have hf₁ : ∃ (v : Set ι),
  Membership.mem (γ := Filter ι) cofinite v ∧
    ∀ y ∈ v, ∀ i ∈ Set.Iic n, Eq (α := R) ((f y : (a : ℕ) → R) i : R) ((1 : (a : ℕ) → R) i : R)
Tactic: rw [← eventually_all_finite (Set.finite_Iic n), eventually_iff_exists_mem] at hf
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
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
hf :
  ∃ (v : Set ι),
    Membership.mem (γ := Filter ι) cofinite v ∧
      ∀ y ∈ v, ∀ i ∈ Set.Iic n, Eq (α := R) ((f y : (a : ℕ) → R) i : R) ((1 : (a : ℕ) → R) i : 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)
have obtain : ∀ (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)),
  ∃ (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: obtain ⟨s, hs, hs'⟩ := hf
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
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)
⊢ ∃ (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)
have refine : ∀ (u v : Finset ι) (hu : t ≤ u) (hv : t ≤ v),
  Eq.{u_2 + 1} (α := R) ((∏ b ∈ u, f b : (a : ℕ) → R) n : R) ((∏ b ∈ v, f b : (a : ℕ) → R) n : R)
Tactic: refine ⟨t, fun u v hu hv ↦ ?_⟩
-- 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 : t ≤ u
hv : t ≤ v
⊢ Eq.{u_2 + 1} (α := R) ((∏ b ∈ u, f b : (a : ℕ) → R) n : R) ((∏ b ∈ v, f b : (a : ℕ) → R) n : R)
have rw : Eq (α := R) ((HMul.hMul (α := ArithmeticFunction R) (∏ x ∈ u \ t, f x) (∏ x ∈ t, f x) : (a : ℕ) → R) n : R)
  ((HMul.hMul (α := ArithmeticFunction R) (∏ x ∈ v \ t, f x) (∏ x ∈ t, f x) : (a : ℕ) → R) n : R)
Tactic: rw [← Finset.prod_sdiff hu, ← Finset.prod_sdiff hv]
-- 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 : t ≤ u
hv : t ≤ v
i : ι
hi : i ∈ u \ t
⊢ i ∈ s
have hi₁ : i ∈ u ∧ i ∈ s
Tactic: rw [Finset.mem_sdiff, Set.Finite.mem_toFinset, Set.notMem_compl_iff] at hi
Cleared hypotheses: hi
-- 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 : t ≤ u
hv : t ≤ v
⊢ Eq (α := R) ((HMul.hMul (α := ArithmeticFunction R) (∏ x ∈ u \ t, f x) (∏ x ∈ t, f x) : (a : ℕ) → R) n : R)
    ((HMul.hMul (α := ArithmeticFunction R) (∏ x ∈ v \ t, f x) (∏ x ∈ t, f x) : (a : ℕ) → R) n : R)
have hu₁ : ∀ i ∈ u \ t, i ∈ s
Tactic: 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
Cleared hypotheses: hu
-- 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
hv : t ≤ v
hu : ∀ i ∈ u \ t, i ∈ s
i : ι
hi : i ∈ v \ t
⊢ i ∈ s
have hi₁ : i ∈ v ∧ i ∈ s
Tactic: rw [Finset.mem_sdiff, Set.Finite.mem_toFinset, Set.notMem_compl_iff] at hi
Cleared hypotheses: hi
-- 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
hv : t ≤ v
hu : ∀ i ∈ u \ t, i ∈ s
⊢ Eq (α := R) ((HMul.hMul (α := ArithmeticFunction R) (∏ x ∈ u \ t, f x) (∏ x ∈ t, f x) : (a : ℕ) → R) n : R)
    ((HMul.hMul (α := ArithmeticFunction R) (∏ x ∈ v \ t, f x) (∏ x ∈ t, f x) : (a : ℕ) → R) n : R)
have hv₁ : ∀ i ∈ v \ t, i ∈ s
Tactic: 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
Cleared hypotheses: hv
-- 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_2 :
  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
this : ∀ k ≤ n, Eq.{u_2 + 1} (α := R) ((∏ x ∈ u \ t, f x : (a : ℕ) → R) k : R) ((∏ x ∈ v \ t, f x : (a : ℕ) → R) k : R)
⊢ Eq (α := R) ((HMul.hMul (α := ArithmeticFunction R) (∏ x ∈ u \ t, f x) (∏ x ∈ t, f x) : (a : ℕ) → R) n : R)
    ((HMul.hMul (α := ArithmeticFunction R) (∏ x ∈ v \ t, f x) (∏ x ∈ t, f x) : (a : ℕ) → R) n : R)
have rw : Eq.{u_2 + 1} (α := R)
  (∑ x ∈ n.divisorsAntidiagonal,
    HMul.hMul (α := R) (β := R) ((∏ x ∈ u \ t, f x : (a : ℕ) → R) x.1 : R) ((∏ x ∈ t, f x : (a : ℕ) → R) x.2 : R))
  (∑ x ∈ n.divisorsAntidiagonal,
    HMul.hMul (α := R) (β := R) ((∏ x ∈ v \ t, f x : (a : ℕ) → R) x.1 : R) ((∏ x ∈ t, f x : (a : ℕ) → R) x.2 : R))
Tactic: rw [mul_apply, mul_apply]
-- 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_2 :
  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
this : ∀ k ≤ n, Eq.{u_2 + 1} (α := R) ((∏ x ∈ u \ t, f x : (a : ℕ) → R) k : R) ((∏ x ∈ v \ t, f x : (a : ℕ) → R) k : R)
⊢ Eq.{u_2 + 1} (α := R)
    (∑ x ∈ n.divisorsAntidiagonal,
      HMul.hMul (α := R) (β := R) ((∏ x ∈ u \ t, f x : (a : ℕ) → R) x.1 : R) ((∏ x ∈ t, f x : (a : ℕ) → R) x.2 : R))
    (∑ x ∈ n.divisorsAntidiagonal,
      HMul.hMul (α := R) (β := R) ((∏ x ∈ v \ t, f x : (a : ℕ) → R) x.1 : R) ((∏ x ∈ t, f x : (a : ℕ) → R) x.2 : R))
have refine : ∀ (k : ℕ × ℕ) (hk : k ∈ n.divisorsAntidiagonal),
  HMul.hMul (α := R) (β := R) ((∏ x ∈ u \ t, f x : (a : ℕ) → R) k.1 : R) ((∏ x ∈ t, f x : (a : ℕ) → R) k.2 : R) =
    HMul.hMul (α := R) (β := R) ((∏ x ∈ v \ t, f x : (a : ℕ) → R) k.1 : R) ((∏ x ∈ t, f x : (a : ℕ) → R) k.2 : R)
Tactic: refine Finset.sum_congr rfl fun k 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_2 :
  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
this : ∀ k ≤ n, Eq.{u_2 + 1} (α := R) ((∏ x ∈ u \ t, f x : (a : ℕ) → R) k : R) ((∏ x ∈ v \ t, f x : (a : ℕ) → R) k : R)
k : ℕ × ℕ
hk : k ∈ n.divisorsAntidiagonal
⊢ HMul.hMul (α := R) (β := R) ((∏ x ∈ u \ t, f x : (a : ℕ) → R) k.1 : R) ((∏ x ∈ t, f x : (a : ℕ) → R) k.2 : R) =
    HMul.hMul (α := R) (β := R) ((∏ x ∈ v \ t, f x : (a : ℕ) → R) k.1 : R) ((∏ x ∈ t, f x : (a : ℕ) → R) k.2 : R)
have rw : HMul.hMul (α := R) (β := R) ((∏ x ∈ v \ t, f x : (a : ℕ) → R) k.1 : R) ((∏ x ∈ t, f x : (a : ℕ) → R) k.2 : R) =
  HMul.hMul (α := R) (β := R) ((∏ x ∈ v \ t, f x : (a : ℕ) → R) k.1 : R) ((∏ x ∈ t, f x : (a : ℕ) → R) k.2 : R)
Tactic: rw [this k.1 (Nat.divisor_le (Nat.fst_mem_divisors_of_mem_antidiagonal 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
⊢ Eq (α := R) ((HMul.hMul (α := ArithmeticFunction R) (∏ x ∈ u \ t, f x) (∏ x ∈ t, f x) : (a : ℕ) → R) n : R)
    ((HMul.hMul (α := ArithmeticFunction R) (∏ x ∈ v \ t, f x) (∏ x ∈ t, f x) : (a : ℕ) → R) n : R)
have suffices : ∀ k ≤ n, Eq.{u_2 + 1} (α := R) ((∏ x ∈ u \ t, f x : (a : ℕ) → R) k : R) ((∏ x ∈ v \ t, f x : (a : ℕ) → R) k : R)
Tactic: 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))]
-- 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_2 :
  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
this :
  ∀ (w : Finset ι),
    (∀ 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)
k : ℕ
hk : k ≤ n
⊢ Eq.{u_2 + 1} (α := R) ((∏ x ∈ u \ t, f x : (a : ℕ) → R) k : R) ((∏ x ∈ v \ t, f x : (a : ℕ) → R) k : R)
have rw : Eq.{u_2 + 1} (α := R) ((1 : (a : ℕ) → R) k : R) ((∏ x ∈ v \ t, f x : (a : ℕ) → R) k : R)
Tactic: rw [this (u \ t) hu k hk, this (v \ t) hv k 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_2 :
  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
this :
  ∀ (w : Finset ι),
    (∀ 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)
k : ℕ
hk : k ≤ n
rw : Eq.{u_2 + 1} (α := R) ((1 : (a : ℕ) → R) k : R) ((∏ x ∈ v \ t, f x : (a : ℕ) → R) k : R)
⊢ Eq.{u_2 + 1} (α := R) ((∏ x ∈ u \ t, f x : (a : ℕ) → R) k : R) ((∏ x ∈ v \ t, f x : (a : ℕ) → R) k : R)
have rw₁ : Eq.{u_2 + 1} (α := R) ((1 : (a : ℕ) → R) k : R) ((1 : (a : ℕ) → R) k : R)
Tactic: rw [this (u \ t) hu k hk, this (v \ t) hv k 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
⊢ ∀ k ≤ n, Eq.{u_2 + 1} (α := R) ((∏ x ∈ u \ t, f x : (a : ℕ) → R) k : R) ((∏ x ∈ v \ t, f x : (a : ℕ) → R) k : R)
have suffices : ∀ (w : Finset ι),
  (∀ 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)
Tactic: 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]
-- 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
w : Finset ι
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)
have induction : ∀ (hw : ∀ (i : ι), Membership.mem (γ := Finset ι) ∅ i → i ∈ s),
  ∀ k ≤ n, Eq.{u_2 + 1} (α := R) ((∏ x ∈ ∅, f x : (a : ℕ) → R) k : R) ((1 : (a : ℕ) → R) k : R)
Tactic: induction w using Finset.induction_on
Cleared hypotheses: w, hw
-- 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
w : Finset ι
hw : ∀ i ∈ w, i ∈ s
induction :
  ∀ (hw : ∀ (i : ι), Membership.mem (γ := Finset ι) ∅ i → i ∈ s),
    ∀ k ≤ n, Eq.{u_2 + 1} (α := R) ((∏ x ∈ ∅, f x : (a : ℕ) → R) k : R) ((1 : (a : ℕ) → R) k : R)
⊢ ∀ k ≤ n, Eq.{u_2 + 1} (α := R) ((∏ x ∈ w, f x : (a : ℕ) → R) k : R) ((1 : (a : ℕ) → R) k : R)
have insert : ∀ (a : ι) (s_1 : Finset ι),
  a ∉ s_1 →
    (∀ (hw : ∀ i ∈ s_1, i ∈ s),
        ∀ k ≤ n, Eq.{u_2 + 1} (α := R) ((∏ x ∈ s_1, f x : (a : ℕ) → R) k : R) ((1 : (a : ℕ) → R) k : R)) →
      ∀ (hw : ∀ i ∈ insert a s_1, i ∈ s),
        ∀ k ≤ n, Eq.{u_2 + 1} (α := R) ((∏ x ∈ insert a s_1, f x : (a : ℕ) → R) k : R) ((1 : (a : ℕ) → R) k : R)
Tactic: induction w using Finset.induction_on
Cleared hypotheses: w, hw
-- 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 ∈ insert i w, f x : (a : ℕ) → R) k : R) ((1 : (a : ℕ) → R) k : R)
have rw : 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))
Tactic: rw [← one_mul (1 : ArithmeticFunction R), Finset.prod_insert hi, mul_apply, mul_apply]