awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Carmichael (goal)
Goal:
n : ℕ
hn : 3 ≤ n
rw : HPow.hPow (α := ℕ) 2 (n - 2).succ = HPow.hPow (α := ℕ) 2 (n - 1) * HSub.hSub (α := ℕ) 2 1
⊢ HMul.hMul (α := ℕ) (β := ℕ) (2 : ℕ) ((λ : (a : ℕ) → ℕ) (HPow.hPow (α := ℕ) 2 n) : ℕ) = φ (HPow.hPow (α := ℕ) 2 n)
have rw₁ : 0 < n
Tactic: rw [carmichael_two_pow_of_ne_two, ← pow_succ', totient_prime_pow prime_two]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Carmichael (goal)
Goal:
n : ℕ
hn : 3 ≤ n
rw : HPow.hPow (α := ℕ) 2 (n - 2).succ = HPow.hPow (α := ℕ) 2 (n - 1) * HSub.hSub (α := ℕ) 2 1
rw₁ : 0 < n
⊢ HMul.hMul (α := ℕ) (β := ℕ) (2 : ℕ) ((λ : (a : ℕ) → ℕ) (HPow.hPow (α := ℕ) 2 n) : ℕ) = φ (HPow.hPow (α := ℕ) 2 n)
have rw₂ : n ≠ 2
Tactic: rw [carmichael_two_pow_of_ne_two, ← pow_succ', totient_prime_pow prime_two]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Carmichael (hypothesis)
Goal:
p n : ℕ
hp : Nat.Prime p
hp₂ : p ≠ 2
⊢ (λ : (a : ℕ) → ℕ) (p ^ n) = φ (p ^ n)
have this : NeZero p
Tactic: have : NeZero p := ⟨hp.ne_zero⟩
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Carmichael (goal)
Goal:
p n : ℕ
hp : Nat.Prime p
hp₂ : p ≠ 2
this : NeZero p
⊢ (λ : (a : ℕ) → ℕ) (p ^ n) = φ (p ^ n)
have rw : exponent (ZMod (p ^ n))ˣ = Nat.card (ZMod (p ^ n))ˣ
Tactic: rw [carmichael_eq_exponent', ← ZMod.card_units_eq_totient, Fintype.card_eq_nat_card]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : Zero R
⊢ Eq.{u_1 + 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.Defs (goal)
Goal:
R : Type u_1
inst : AddGroupWithOne R
f : ArithmeticFunction ℕ
⊢ Eq.{u_1 + 1} (α := ArithmeticFunction R) (↑(↑f : ArithmeticFunction ℤ) : ArithmeticFunction R)
    (↑f : ArithmeticFunction R)
have h : ∀ (x : ℕ),
  Eq.{u_1 + 1} (α := R) (((↑(↑f : ArithmeticFunction ℤ) : ArithmeticFunction R) : (a : ℕ) → R) x : R)
    (((↑f : ArithmeticFunction R) : (a : ℕ) → R) x : R)
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : AddMonoidWithOne R
⊢ Eq.{u_1 + 1} (α := ArithmeticFunction R) (↑1 : ArithmeticFunction R) 1
have h : ∀ (n : ℕ), Eq.{u_1 + 1} (α := R) (((↑1 : ArithmeticFunction R) : (a : ℕ) → R) n : R) ((1 : (a : ℕ) → R) n : R)
Tactic: ext n
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : AddGroupWithOne R
⊢ Eq.{u_1 + 1} (α := ArithmeticFunction R) (↑1 : ArithmeticFunction R) 1
have h : ∀ (n : ℕ), Eq.{u_1 + 1} (α := R) (((↑1 : ArithmeticFunction R) : (a : ℕ) → R) n : R) ((1 : (a : ℕ) → R) n : R)
Tactic: ext n
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : AddMonoidWithOne R
⊢ Eq (α := ArithmeticFunction R)
    ({ toFun := fun (x : ℕ) ↦ if x = 1 then (↑0 : R) else 0, map_zero' := ⋯ } : ZeroHom ℕ R) (0 : ArithmeticFunction R)
have h : ∀ (x : ℕ),
  Eq (α := R)
    ((({ toFun := fun (x : ℕ) ↦ if x = 1 then (↑0 : R) else 0, map_zero' := sorry } : ZeroHom ℕ R) : (a : ℕ) → R) x : R)
    ((0 : (a : ℕ) → R) x : R)
Tactic: ext
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : AddMonoidWithOne R
n : ℕ
⊢ { toFun := fun (x : ℕ) ↦ if x = 1 then (↑(n + 1) : R) else 0, map_zero' := ⋯ } =
    HAdd.hAdd (α := ArithmeticFunction R)
      { toFun := fun (x : ℕ) ↦ if x = 1 then (↑n : R) else 0, map_zero' := ⋯ : ZeroHom ℕ R } 1
have h : ∀ (x : ℕ),
  Eq (α := R)
    ((({ toFun := fun (x : ℕ) ↦ if x = 1 then (↑(n + 1) : R) else 0, map_zero' := sorry } : ZeroHom ℕ R) : (a : ℕ) → R) x :
      R)
    ((HAdd.hAdd (α := ArithmeticFunction R)
            { toFun := fun (x : ℕ) ↦ if x = 1 then (↑n : R) else 0, map_zero' := sorry : ZeroHom ℕ R } 1 :
          (a : ℕ) → R)
        x :
      R)
Tactic: ext x
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : AddMonoidWithOne R
n x : ℕ
⊢ Eq (α := R)
    ((({ toFun := fun (x : ℕ) ↦ if x = 1 then (↑(n + 1) : R) else 0, map_zero' := ⋯ } : ZeroHom ℕ R) : (a : ℕ) → R) x :
      R)
    ((HAdd.hAdd (α := ArithmeticFunction R)
            { toFun := fun (x : ℕ) ↦ if x = 1 then (↑n : R) else 0, map_zero' := ⋯ : ZeroHom ℕ R } 1 :
          (a : ℕ) → R)
        x :
      R)
have pos : ∀ (h : x = 1),
  Eq (α := R)
    ((({ toFun := fun (x : ℕ) ↦ if x = 1 then (↑(n + 1) : R) else 0, map_zero' := sorry } : ZeroHom ℕ R) : (a : ℕ) → R) x :
      R)
    ((HAdd.hAdd (α := ArithmeticFunction R)
            { toFun := fun (x : ℕ) ↦ if x = 1 then (↑n : R) else 0, map_zero' := sorry : ZeroHom ℕ R } 1 :
          (a : ℕ) → R)
        x :
      R)
Tactic: by_cases h : x = 1
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : AddMonoidWithOne R
n x : ℕ
pos :
  ∀ (h : x = 1),
    Eq (α := R)
      ((({ toFun := fun (x : ℕ) ↦ if x = 1 then (↑(n + 1) : R) else 0, map_zero' := ⋯ } : ZeroHom ℕ R) : (a : ℕ) → R)
          x :
        R)
      ((HAdd.hAdd (α := ArithmeticFunction R)
              { toFun := fun (x : ℕ) ↦ if x = 1 then (↑n : R) else 0, map_zero' := ⋯ : ZeroHom ℕ R } 1 :
            (a : ℕ) → R)
          x :
        R)
⊢ Eq (α := R)
    ((({ toFun := fun (x : ℕ) ↦ if x = 1 then (↑(n + 1) : R) else 0, map_zero' := ⋯ } : ZeroHom ℕ R) : (a : ℕ) → R) x :
      R)
    ((HAdd.hAdd (α := ArithmeticFunction R)
            { toFun := fun (x : ℕ) ↦ if x = 1 then (↑n : R) else 0, map_zero' := ⋯ : ZeroHom ℕ R } 1 :
          (a : ℕ) → R)
        x :
      R)
have neg : ∀ (h : ¬x = 1),
  Eq (α := R)
    ((({ toFun := fun (x : ℕ) ↦ if x = 1 then (↑(n + 1) : R) else 0, map_zero' := sorry } : ZeroHom ℕ R) : (a : ℕ) → R) x :
      R)
    ((HAdd.hAdd (α := ArithmeticFunction R)
            { toFun := fun (x : ℕ) ↦ if x = 1 then (↑n : R) else 0, map_zero' := sorry : ZeroHom ℕ R } 1 :
          (a : ℕ) → R)
        x :
      R)
Tactic: by_cases h : x = 1
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : Semiring R
f g : ArithmeticFunction ℕ
⊢ (↑(f * g) : ArithmeticFunction R) =
    HMul.hMul (α := ArithmeticFunction R) (↑f : ArithmeticFunction R) (↑g : ArithmeticFunction R)
have h : ∀ (n : ℕ),
  Eq (α := R) (((↑(f * g) : ArithmeticFunction R) : (a : ℕ) → R) n : R)
    ((HMul.hMul (α := ArithmeticFunction R) (↑f : ArithmeticFunction R) (↑g : ArithmeticFunction R) : (a : ℕ) → R) n :
      R)
Tactic: ext n
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : Ring R
f g : ArithmeticFunction ℤ
⊢ (↑(f * g) : ArithmeticFunction R) =
    HMul.hMul (α := ArithmeticFunction R) (↑f : ArithmeticFunction R) (↑g : ArithmeticFunction R)
have h : ∀ (n : ℕ),
  Eq (α := R) (((↑(f * g) : ArithmeticFunction R) : (a : ℕ) → R) n : R)
    ((HMul.hMul (α := ArithmeticFunction R) (↑f : ArithmeticFunction R) (↑g : ArithmeticFunction R) : (a : ℕ) → R) n :
      R)
Tactic: ext n
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
M : Type u_2
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
f g : ArithmeticFunction R
h : ArithmeticFunction M
⊢ (f * g) • h = f • g • h
have h₁ : ∀ (n : ℕ), Eq (α := M) (((f * g) • h : (a : ℕ) → M) n : M) ((f • g • h : (a : ℕ) → M) n : M)
Tactic: ext n
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
M : Type u_2
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
f g : ArithmeticFunction R
h : ArithmeticFunction M
n : ℕ
⊢ Eq (α := M) (((f * g) • h : (a : ℕ) → M) n : M) ((f • g • h : (a : ℕ) → M) n : M)
have h₁ : Eq.{u_2 + 1} (α := M)
  (∑ x ∈ n.divisorsAntidiagonal.sigma (α := fun (a : ℕ × ℕ) ↦ ℕ × ℕ) fun (a : ℕ × ℕ) ↦ a.1.divisorsAntidiagonal,
    HSMul.hSMul (α := R) ((f : (a : ℕ) → R) x.snd (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ).1 : R)
      (HSMul.hSMul (α := R) (β := M) ((g : (a : ℕ) → R) x.snd (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ).2 : R)
        ((h : (a : ℕ) → M) x.fst (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ).2 : M)))
  (∑ x ∈ n.divisorsAntidiagonal.sigma (α := fun (a : ℕ × ℕ) ↦ ℕ × ℕ) fun (a : ℕ × ℕ) ↦ a.2.divisorsAntidiagonal,
    HSMul.hSMul (α := R) ((f : (a : ℕ) → R) x.fst (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ).1 : R)
      (HSMul.hSMul (α := R) (β := M) ((g : (a : ℕ) → R) x.snd (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ).1 : R)
        ((h : (a : ℕ) → M) x.snd (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ).2 : M)))
Tactic: simp only [mul_apply, smul_apply, sum_smul, mul_smul, smul_sum, sum_sigma']
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
M : Type u_2
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
f g : ArithmeticFunction R
h : ArithmeticFunction M
n : ℕ
⊢ Eq.{u_2 + 1} (α := M)
    (∑ x ∈ n.divisorsAntidiagonal.sigma (α := fun (a : ℕ × ℕ) ↦ ℕ × ℕ) fun (a : ℕ × ℕ) ↦ a.1.divisorsAntidiagonal,
      HSMul.hSMul (α := R) ((f : (a : ℕ) → R) x.snd (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ).1 : R)
        (HSMul.hSMul (α := R) (β := M) ((g : (a : ℕ) → R) x.snd (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ).2 : R)
          ((h : (a : ℕ) → M) x.fst (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ).2 : M)))
    (∑ x ∈ n.divisorsAntidiagonal.sigma (α := fun (a : ℕ × ℕ) ↦ ℕ × ℕ) fun (a : ℕ × ℕ) ↦ a.2.divisorsAntidiagonal,
      HSMul.hSMul (α := R) ((f : (a : ℕ) → R) x.fst (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ).1 : R)
        (HSMul.hSMul (α := R) (β := M) ((g : (a : ℕ) → R) x.snd (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ).1 : R)
          ((h : (a : ℕ) → M) x.snd (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ).2 : M)))
have h₁ : ∀ (a : (_ : ℕ × ℕ) × ℕ × ℕ),
  Membership.mem (γ := Finset ((_ : ℕ × ℕ) × ℕ × ℕ))
      (n.divisorsAntidiagonal.sigma (α := fun (a : ℕ × ℕ) ↦ ℕ × ℕ) fun (a : ℕ × ℕ) ↦ a.1.divisorsAntidiagonal) a →
    Membership.mem (α := (_ : ℕ × ℕ) × ℕ × ℕ) (γ := Finset ((_ : ℕ × ℕ) × ℕ × ℕ))
      (n.divisorsAntidiagonal.sigma (α := fun (a : ℕ × ℕ) ↦ ℕ × ℕ) fun (a : ℕ × ℕ) ↦ a.2.divisorsAntidiagonal)
      (match (motive := (_ : ℕ × ℕ) × ℕ × ℕ → (_ : ℕ × ℕ) × ℕ × ℕ) a with
        | ⟨(_i, j), (k, l)⟩ => Sigma.mk (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ) (k, l * j) (l, j) :
        (_ : ℕ × ℕ) × ℕ × ℕ)
Tactic: apply sum_nbij' (fun ⟨⟨_i, j⟩, ⟨k, l⟩⟩ ↦ ⟨(k, l * j), (l, j)⟩)
    (fun ⟨⟨i, _j⟩, ⟨k, l⟩⟩ ↦ ⟨(i * k, l), (i, k)⟩)
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
M : Type u_2
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
f g : ArithmeticFunction R
h : ArithmeticFunction M
n : ℕ
h₁ :
  ∀ (a : (_ : ℕ × ℕ) × ℕ × ℕ),
    Membership.mem (γ := Finset ((_ : ℕ × ℕ) × ℕ × ℕ))
        (n.divisorsAntidiagonal.sigma (α := fun (a : ℕ × ℕ) ↦ ℕ × ℕ) fun (a : ℕ × ℕ) ↦ a.1.divisorsAntidiagonal) a →
      Membership.mem (α := (_ : ℕ × ℕ) × ℕ × ℕ) (γ := Finset ((_ : ℕ × ℕ) × ℕ × ℕ))
        (n.divisorsAntidiagonal.sigma (α := fun (a : ℕ × ℕ) ↦ ℕ × ℕ) fun (a : ℕ × ℕ) ↦ a.2.divisorsAntidiagonal)
        (match (motive := (_ : ℕ × ℕ) × ℕ × ℕ → (_ : ℕ × ℕ) × ℕ × ℕ) a with
          | ⟨(_i, j), (k, l)⟩ => Sigma.mk (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ) (k, l * j) (l, j) :
          (_ : ℕ × ℕ) × ℕ × ℕ)
⊢ Eq.{u_2 + 1} (α := M)
    (∑ x ∈ n.divisorsAntidiagonal.sigma (α := fun (a : ℕ × ℕ) ↦ ℕ × ℕ) fun (a : ℕ × ℕ) ↦ a.1.divisorsAntidiagonal,
      HSMul.hSMul (α := R) ((f : (a : ℕ) → R) x.snd (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ).1 : R)
        (HSMul.hSMul (α := R) (β := M) ((g : (a : ℕ) → R) x.snd (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ).2 : R)
          ((h : (a : ℕ) → M) x.fst (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ).2 : M)))
    (∑ x ∈ n.divisorsAntidiagonal.sigma (α := fun (a : ℕ × ℕ) ↦ ℕ × ℕ) fun (a : ℕ × ℕ) ↦ a.2.divisorsAntidiagonal,
      HSMul.hSMul (α := R) ((f : (a : ℕ) → R) x.fst (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ).1 : R)
        (HSMul.hSMul (α := R) (β := M) ((g : (a : ℕ) → R) x.snd (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ).1 : R)
          ((h : (a : ℕ) → M) x.snd (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ).2 : M)))
have h₂ : ∀ (a : (_ : ℕ × ℕ) × ℕ × ℕ),
  Membership.mem (γ := Finset ((_ : ℕ × ℕ) × ℕ × ℕ))
      (n.divisorsAntidiagonal.sigma (α := fun (a : ℕ × ℕ) ↦ ℕ × ℕ) fun (a : ℕ × ℕ) ↦ a.2.divisorsAntidiagonal) a →
    Membership.mem (α := (_ : ℕ × ℕ) × ℕ × ℕ) (γ := Finset ((_ : ℕ × ℕ) × ℕ × ℕ))
      (n.divisorsAntidiagonal.sigma (α := fun (a : ℕ × ℕ) ↦ ℕ × ℕ) fun (a : ℕ × ℕ) ↦ a.1.divisorsAntidiagonal)
      (match (motive := (_ : ℕ × ℕ) × ℕ × ℕ → (_ : ℕ × ℕ) × ℕ × ℕ) a with
        | ⟨(i, _j), (k, l)⟩ => Sigma.mk (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ) (i * k, l) (i, k) :
        (_ : ℕ × ℕ) × ℕ × ℕ)
Tactic: apply sum_nbij' (fun ⟨⟨_i, j⟩, ⟨k, l⟩⟩ ↦ ⟨(k, l * j), (l, j)⟩)
    (fun ⟨⟨i, _j⟩, ⟨k, l⟩⟩ ↦ ⟨(i * k, l), (i, k)⟩)
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
M : Type u_2
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
f g : ArithmeticFunction R
h : ArithmeticFunction M
n : ℕ
h₁ :
  ∀ (a : (_ : ℕ × ℕ) × ℕ × ℕ),
    Membership.mem (γ := Finset ((_ : ℕ × ℕ) × ℕ × ℕ))
        (n.divisorsAntidiagonal.sigma (α := fun (a : ℕ × ℕ) ↦ ℕ × ℕ) fun (a : ℕ × ℕ) ↦ a.1.divisorsAntidiagonal) a →
      Membership.mem (α := (_ : ℕ × ℕ) × ℕ × ℕ) (γ := Finset ((_ : ℕ × ℕ) × ℕ × ℕ))
        (n.divisorsAntidiagonal.sigma (α := fun (a : ℕ × ℕ) ↦ ℕ × ℕ) fun (a : ℕ × ℕ) ↦ a.2.divisorsAntidiagonal)
        (match (motive := (_ : ℕ × ℕ) × ℕ × ℕ → (_ : ℕ × ℕ) × ℕ × ℕ) a with
          | ⟨(_i, j), (k, l)⟩ => Sigma.mk (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ) (k, l * j) (l, j) :
          (_ : ℕ × ℕ) × ℕ × ℕ)
h₂ :
  ∀ (a : (_ : ℕ × ℕ) × ℕ × ℕ),
    Membership.mem (γ := Finset ((_ : ℕ × ℕ) × ℕ × ℕ))
        (n.divisorsAntidiagonal.sigma (α := fun (a : ℕ × ℕ) ↦ ℕ × ℕ) fun (a : ℕ × ℕ) ↦ a.2.divisorsAntidiagonal) a →
      Membership.mem (α := (_ : ℕ × ℕ) × ℕ × ℕ) (γ := Finset ((_ : ℕ × ℕ) × ℕ × ℕ))
        (n.divisorsAntidiagonal.sigma (α := fun (a : ℕ × ℕ) ↦ ℕ × ℕ) fun (a : ℕ × ℕ) ↦ a.1.divisorsAntidiagonal)
        (match (motive := (_ : ℕ × ℕ) × ℕ × ℕ → (_ : ℕ × ℕ) × ℕ × ℕ) a with
          | ⟨(i, _j), (k, l)⟩ => Sigma.mk (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ) (i * k, l) (i, k) :
          (_ : ℕ × ℕ) × ℕ × ℕ)
⊢ Eq.{u_2 + 1} (α := M)
    (∑ x ∈ n.divisorsAntidiagonal.sigma (α := fun (a : ℕ × ℕ) ↦ ℕ × ℕ) fun (a : ℕ × ℕ) ↦ a.1.divisorsAntidiagonal,
      HSMul.hSMul (α := R) ((f : (a : ℕ) → R) x.snd (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ).1 : R)
        (HSMul.hSMul (α := R) (β := M) ((g : (a : ℕ) → R) x.snd (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ).2 : R)
          ((h : (a : ℕ) → M) x.fst (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ).2 : M)))
    (∑ x ∈ n.divisorsAntidiagonal.sigma (α := fun (a : ℕ × ℕ) ↦ ℕ × ℕ) fun (a : ℕ × ℕ) ↦ a.2.divisorsAntidiagonal,
      HSMul.hSMul (α := R) ((f : (a : ℕ) → R) x.fst (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ).1 : R)
        (HSMul.hSMul (α := R) (β := M) ((g : (a : ℕ) → R) x.snd (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ).1 : R)
          ((h : (a : ℕ) → M) x.snd (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ).2 : M)))
have h₃ : ∀ (a : (_ : ℕ × ℕ) × ℕ × ℕ),
  Membership.mem (γ := Finset ((_ : ℕ × ℕ) × ℕ × ℕ))
      (n.divisorsAntidiagonal.sigma (α := fun (a : ℕ × ℕ) ↦ ℕ × ℕ) fun (a : ℕ × ℕ) ↦ a.1.divisorsAntidiagonal) a →
    (match (motive := (_ : ℕ × ℕ) × ℕ × ℕ → (_ : ℕ × ℕ) × ℕ × ℕ)
        match (motive := (_ : ℕ × ℕ) × ℕ × ℕ → (_ : ℕ × ℕ) × ℕ × ℕ) a with
        | ⟨(_i, j), (k, l)⟩ => Sigma.mk (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ) (k, l * j) (l, j) with
      | ⟨(i, _j), (k, l)⟩ => Sigma.mk (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ) (i * k, l) (i, k)) =
      a
Tactic: apply sum_nbij' (fun ⟨⟨_i, j⟩, ⟨k, l⟩⟩ ↦ ⟨(k, l * j), (l, j)⟩)
    (fun ⟨⟨i, _j⟩, ⟨k, l⟩⟩ ↦ ⟨(i * k, l), (i, k)⟩)
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
M : Type u_2
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
b : ArithmeticFunction M
⊢ HSMul.hSMul.{u_1, u_2, u_2} (α := ArithmeticFunction R) 1 b = b
have h : ∀ (x : ℕ),
  Eq (α := M) ((HSMul.hSMul.{u_1, u_2, u_2} (α := ArithmeticFunction R) 1 b : (a : ℕ) → M) x : M)
    ((b : (a : ℕ) → M) x : M)
Tactic: ext x
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
M : Type u_2
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
b : ArithmeticFunction M
x : ℕ
⊢ Eq (α := M) ((HSMul.hSMul.{u_1, u_2, u_2} (α := ArithmeticFunction R) 1 b : (a : ℕ) → M) x : M)
    ((b : (a : ℕ) → M) x : M)
have h : Eq (α := M)
  (∑ x ∈ x.divisorsAntidiagonal,
      HSMul.hSMul.{u_1, u_2, u_2} (α := R) (β := M) ((1 : (a : ℕ) → R) x.1 : R) ((b : (a : ℕ) → M) x.2 : M) :
    M)
  ((b : (a : ℕ) → M) x : M)
Tactic: rw [smul_apply]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
M : Type u_2
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
b : ArithmeticFunction M
x : ℕ
⊢ Eq (α := M)
    (∑ x ∈ x.divisorsAntidiagonal,
        HSMul.hSMul.{u_1, u_2, u_2} (α := R) (β := M) ((1 : (a : ℕ) → R) x.1 : R) ((b : (a : ℕ) → M) x.2 : M) :
      M)
    ((b : (a : ℕ) → M) x : M)
have pos : ∀ (x0 : x = 0),
  Eq (α := M)
    (∑ x ∈ x.divisorsAntidiagonal,
        HSMul.hSMul.{u_1, u_2, u_2} (α := R) (β := M) ((1 : (a : ℕ) → R) x.1 : R) ((b : (a : ℕ) → M) x.2 : M) :
      M)
    ((b : (a : ℕ) → M) x : M)
Tactic: by_cases x0 : x = 0
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
M : Type u_2
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
b : ArithmeticFunction M
x : ℕ
pos :
  ∀ (x0 : x = 0),
    Eq (α := M)
      (∑ x ∈ x.divisorsAntidiagonal,
          HSMul.hSMul.{u_1, u_2, u_2} (α := R) (β := M) ((1 : (a : ℕ) → R) x.1 : R) ((b : (a : ℕ) → M) x.2 : M) :
        M)
      ((b : (a : ℕ) → M) x : M)
⊢ Eq (α := M)
    (∑ x ∈ x.divisorsAntidiagonal,
        HSMul.hSMul.{u_1, u_2, u_2} (α := R) (β := M) ((1 : (a : ℕ) → R) x.1 : R) ((b : (a : ℕ) → M) x.2 : M) :
      M)
    ((b : (a : ℕ) → M) x : M)
have neg : ∀ (x0 : ¬x = 0),
  Eq (α := M)
    (∑ x ∈ x.divisorsAntidiagonal,
        HSMul.hSMul.{u_1, u_2, u_2} (α := R) (β := M) ((1 : (a : ℕ) → R) x.1 : R) ((b : (a : ℕ) → M) x.2 : M) :
      M)
    ((b : (a : ℕ) → M) x : M)
Tactic: by_cases x0 : x = 0
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (hypothesis)
Goal:
R : Type u_1
M : Type u_2
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
b : ArithmeticFunction M
x : ℕ
x0 : ¬x = 0
⊢ Eq (α := M)
    (∑ x ∈ x.divisorsAntidiagonal,
        HSMul.hSMul.{u_1, u_2, u_2} (α := R) (β := M) ((1 : (a : ℕ) → R) x.1 : R) ((b : (a : ℕ) → M) x.2 : M) :
      M)
    ((b : (a : ℕ) → M) x : M)
have h : {(1, x)} ⊆ x.divisorsAntidiagonal
Tactic: have h : {(1, x)} ⊆ divisorsAntidiagonal x := by simp [x0]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
M : Type u_2
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
b : ArithmeticFunction M
x : ℕ
x0 : ¬x = 0
h : {(1, x)} ⊆ x.divisorsAntidiagonal
⊢ Eq (α := M)
    (∑ x ∈ x.divisorsAntidiagonal,
        HSMul.hSMul.{u_1, u_2, u_2} (α := R) (β := M) ((1 : (a : ℕ) → R) x.1 : R) ((b : (a : ℕ) → M) x.2 : M) :
      M)
    ((b : (a : ℕ) → M) x : M)
have neg : Eq (α := M)
  (∑ x ∈ {(1, x)},
      HSMul.hSMul.{u_1, u_2, u_2} (α := R) (β := M) ((1 : (a : ℕ) → R) x.1 : R) ((b : (a : ℕ) → M) x.2 : M) :
    M)
  ((b : (a : ℕ) → M) x : M)
Tactic: rw [← sum_subset h]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
M : Type u_2
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
b : ArithmeticFunction M
x : ℕ
x0 : ¬x = 0
h : {(1, x)} ⊆ x.divisorsAntidiagonal
neg :
  Eq (α := M)
    (∑ x ∈ {(1, x)},
        HSMul.hSMul.{u_1, u_2, u_2} (α := R) (β := M) ((1 : (a : ℕ) → R) x.1 : R) ((b : (a : ℕ) → M) x.2 : M) :
      M)
    ((b : (a : ℕ) → M) x : M)
⊢ Eq (α := M)
    (∑ x ∈ x.divisorsAntidiagonal,
        HSMul.hSMul.{u_1, u_2, u_2} (α := R) (β := M) ((1 : (a : ℕ) → R) x.1 : R) ((b : (a : ℕ) → M) x.2 : M) :
      M)
    ((b : (a : ℕ) → M) x : M)
have neg₁ : ∀ x_1 ∈ x.divisorsAntidiagonal,
  x_1 ∉ {(1, x)} →
    HSMul.hSMul.{u_1, u_2, u_2} (α := R) (β := M) ((1 : (a : ℕ) → R) x_1.1 : R) ((b : (a : ℕ) → M) x_1.2 : M) = 0
Tactic: rw [← sum_subset h]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (hypothesis)
Goal:
R : Type u_1
M : Type u_2
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
b : ArithmeticFunction M
x : ℕ
x0 : ¬x = 0
h : {(1, x)} ⊆ x.divisorsAntidiagonal
y : ℕ × ℕ
ymem : y ∈ x.divisorsAntidiagonal
ynotMem : y ∉ {(1, x)}
⊢ HSMul.hSMul.{u_1, u_2, u_2} (α := R) (β := M) ((1 : (a : ℕ) → R) y.1 : R) ((b : (a : ℕ) → M) y.2 : M) = 0
have y1ne : y.1 ≠ 1
Tactic: have y1ne : y.fst ≠ 1 := fun con => by simp_all [Prod.ext_iff]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : Semiring R
f : ArithmeticFunction R
⊢ f * 1 = f
have h : ∀ (x : ℕ), Eq (α := R) ((f * 1 : (a : ℕ) → R) x : R) ((f : (a : ℕ) → R) x : R)
Tactic: ext x
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : Semiring R
f : ArithmeticFunction R
x : ℕ
⊢ Eq (α := R) ((f * 1 : (a : ℕ) → R) x : R) ((f : (a : ℕ) → R) x : R)
have h : Eq (α := R)
  (∑ x ∈ x.divisorsAntidiagonal, HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) x.1 : R) ((1 : (a : ℕ) → R) x.2 : R) :
    R)
  ((f : (a : ℕ) → R) x : R)
Tactic: rw [mul_apply]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : Semiring R
f : ArithmeticFunction R
x : ℕ
⊢ Eq (α := R)
    (∑ x ∈ x.divisorsAntidiagonal, HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) x.1 : R) ((1 : (a : ℕ) → R) x.2 : R) :
      R)
    ((f : (a : ℕ) → R) x : R)
have pos : ∀ (x0 : x = 0),
  Eq (α := R)
    (∑ x ∈ x.divisorsAntidiagonal, HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) x.1 : R) ((1 : (a : ℕ) → R) x.2 : R) :
      R)
    ((f : (a : ℕ) → R) x : R)
Tactic: by_cases x0 : x = 0
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : Semiring R
f : ArithmeticFunction R
x : ℕ
pos :
  ∀ (x0 : x = 0),
    Eq (α := R)
      (∑ x ∈ x.divisorsAntidiagonal,
          HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) x.1 : R) ((1 : (a : ℕ) → R) x.2 : R) :
        R)
      ((f : (a : ℕ) → R) x : R)
⊢ Eq (α := R)
    (∑ x ∈ x.divisorsAntidiagonal, HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) x.1 : R) ((1 : (a : ℕ) → R) x.2 : R) :
      R)
    ((f : (a : ℕ) → R) x : R)
have neg : ∀ (x0 : ¬x = 0),
  Eq (α := R)
    (∑ x ∈ x.divisorsAntidiagonal, HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) x.1 : R) ((1 : (a : ℕ) → R) x.2 : R) :
      R)
    ((f : (a : ℕ) → R) x : R)
Tactic: by_cases x0 : x = 0
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (hypothesis)
Goal:
R : Type u_1
inst : Semiring R
f : ArithmeticFunction R
x : ℕ
x0 : ¬x = 0
⊢ Eq (α := R)
    (∑ x ∈ x.divisorsAntidiagonal, HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) x.1 : R) ((1 : (a : ℕ) → R) x.2 : R) :
      R)
    ((f : (a : ℕ) → R) x : R)
have h : {(x, 1)} ⊆ x.divisorsAntidiagonal
Tactic: have h : {(x, 1)} ⊆ divisorsAntidiagonal x := by simp [x0]
-- Lean 4 proof step from Mathlib.NumberTheory.ArithmeticFunction.Defs (goal)
Goal:
R : Type u_1
inst : Semiring R
f : ArithmeticFunction R
x : ℕ
x0 : ¬x = 0
h : {(x, 1)} ⊆ x.divisorsAntidiagonal
⊢ Eq (α := R)
    (∑ x ∈ x.divisorsAntidiagonal, HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) x.1 : R) ((1 : (a : ℕ) → R) x.2 : R) :
      R)
    ((f : (a : ℕ) → R) x : R)
have neg : Eq (α := R) (∑ x ∈ {(x, 1)}, HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) x.1 : R) ((1 : (a : ℕ) → R) x.2 : R) : R)
  ((f : (a : ℕ) → R) x : R)
Tactic: rw [← sum_subset h]