awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction
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)
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))
⊢ 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 [mul_apply, mul_apply]

Premises used: ArithmeticFunction.mul_apply
-- Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction
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))

Tactic:
Finset.sum_congr rfl fun k hk ↦ ?_

Premises used: Finset.sum_congr, rfl
-- Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction
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)
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)
⊢ 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:
refine Finset.sum_congr rfl fun k hk ↦ ?_

Premises used: Finset.sum_congr, rfl
-- Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction
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
rw rw₁ :
  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)
⊢ 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:
rw [this k.1 (Nat.divisor_le (Nat.fst_mem_divisors_of_mem_antidiagonal hk))]

Premises used: Prod.fst, Nat.fst_mem_divisors_of_mem_antidiagonal, Nat.divisor_le
-- Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction
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
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)
⊢ 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:
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))]
-- Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction
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)
rw₁ rw₂ : Eq.{u_2 + 1} (α := R) ((1 : (a : ℕ) → R) k : R) ((1 : (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)

Tactic:
rw [this (u \ t) hu k hk, this (v \ t) hv k hk]

Premises used: SDiff.sdiff
-- Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction
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
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)
⊢ ∀ 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 ∀ 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]
-- Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction
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)
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.insert a s_1, i ∈ s),
          ∀ k ≤ n,
            Eq.{u_2 + 1} (α := R) ((∏ x ∈ Insert.insert a s_1, 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)

Tactic:
induction w using Finset.induction_on
-- Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction
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
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:
simp
-- Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction
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
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:
case empty => simp
-- Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction
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
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))
⊢ Eq.{u_2 + 1} (α := R) ((∏ x ∈ insert i w, f x : (a : ℕ) → R) k : R) ((1 : (a : ℕ) → R) k : R)

Tactic:
rw [← one_mul (1 : ArithmeticFunction R), Finset.prod_insert hi, mul_apply, mul_apply]

Premises used: Finset.prod_insert, ArithmeticFunction.mul_apply, OfNat.ofNat, one_mul
-- Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction
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))

Tactic:
Finset.sum_congr rfl fun j hj ↦ ?_

Premises used: Finset.sum_congr, rfl
-- Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction
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
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)
⊢ 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:
refine Finset.sum_congr rfl fun j hj ↦ ?_

Premises used: Finset.sum_congr, rfl
-- Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction
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
⊢ 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)
-- Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction
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)
⊢ 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)
-- Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction
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)
rw₁ rw₂ :
  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)
⊢ 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:
rw [h1, h2]
-- Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction
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)
neg : ∀ (hf' : ¬Multipliable f), IsMultiplicative.{u_2} (R := R) (eulerProduct f)
⊢ IsMultiplicative.{u_2} (R := R) (eulerProduct f)

Tactic:
by_cases hf' : Multipliable f

Premises used: Multipliable, SummationFilter.unconditional
-- Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction
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
⊢ ∀ (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
-- Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction
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)
⊢ ∀ (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
-- Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction
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)

Tactic:
(forall_and.mp h).imp (fun h ↦ ?_) fun h m n hmn ↦ ?_

Premises used: And.imp, ArithmeticFunction.IsMultiplicative, Iff.mp, forall_and
-- Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction
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)
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)
⊢ IsMultiplicative.{u_2} (R := R) (eulerProduct f)

Tactic:
refine (forall_and.mp h).imp (fun h ↦ ?_) fun h m n hmn ↦ ?_

Premises used: And.imp, Iff.mp, forall_and
-- Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction
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)
⊢ ∀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

Premises used: OfNat.ofNat
-- Module: Mathlib.NumberTheory.ArithmeticFunction.LFunction
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)
⊢ ∀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