awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
Goal:
R : Type u_1
inst : AddMonoidWithOne R
f : ArithmeticFunction ℕ
x : ℕ
⊢ Eq.{u_1 + 1} (α := R) (((↑f : ArithmeticFunction R) : (a : ℕ) → R) x : R) ((↑((f : (a : ℕ) → ℕ) x) : R) : R)

Tactic:
rfl

Premises used: rfl
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
Goal:
R : Type u_1
inst : AddGroupWithOne R
f : ArithmeticFunction ℤ
⊢ Eq.{u_1 + 1} (α := R) ((fun (n : ℕ) ↦ ((↑((f : (a : ℕ) → ℤ) n) : R) : R)) 0) 0

Tactic:
simp
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
Goal:
f : ArithmeticFunction ℤ
⊢ (↑f : ArithmeticFunction ℤ) = f

Tactic:
ext fun _ => Int.cast_id

Premises used: ArithmeticFunction.ext, Int.cast_id
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
Goal:
R : Type u_1
inst : AddGroupWithOne R
f : ArithmeticFunction ℤ
x : ℕ
⊢ Eq.{u_1 + 1} (α := R) (((↑f : ArithmeticFunction R) : (a : ℕ) → R) x : R) ((↑((f : (a : ℕ) → ℤ) x) : R) : R)

Tactic:
rfl

Premises used: rfl
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
Goal:
R : Type u_1
inst : AddGroupWithOne R
f : ArithmeticFunction ℕ
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)
⊢ Eq.{u_1 + 1} (α := ArithmeticFunction R) (↑(↑f : ArithmeticFunction ℤ) : ArithmeticFunction R)
    (↑f : ArithmeticFunction R)

Tactic:
ext
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
Goal:
R : Type u_1
inst : AddGroupWithOne R
f : ArithmeticFunction ℕ
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:
simp
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
Goal:
R : Type u_1
inst : AddMonoidWithOne R
h : ∀ (n : ℕ), Eq.{u_1 + 1} (α := R) (((↑1 : ArithmeticFunction R) : (a : ℕ) → R) n : R) ((1 : (a : ℕ) → R) n : R)
⊢ Eq.{u_1 + 1} (α := ArithmeticFunction R) (↑1 : ArithmeticFunction R) 1

Tactic:
ext n
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
Goal:
R : Type u_1
inst : AddMonoidWithOne R
n : ℕ
⊢ Eq.{u_1 + 1} (α := R) (((↑1 : ArithmeticFunction R) : (a : ℕ) → R) n : R) ((1 : (a : ℕ) → R) n : R)

Tactic:
simp [one_apply]

Premises used: ArithmeticFunction.one_apply
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
Goal:
R : Type u_1
inst : AddGroupWithOne R
h : ∀ (n : ℕ), Eq.{u_1 + 1} (α := R) (((↑1 : ArithmeticFunction R) : (a : ℕ) → R) n : R) ((1 : (a : ℕ) → R) n : R)
⊢ Eq.{u_1 + 1} (α := ArithmeticFunction R) (↑1 : ArithmeticFunction R) 1

Tactic:
ext n
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
Goal:
R : Type u_1
inst : AddGroupWithOne R
n : ℕ
⊢ Eq.{u_1 + 1} (α := R) (((↑1 : ArithmeticFunction R) : (a : ℕ) → R) n : R) ((1 : (a : ℕ) → R) n : R)

Tactic:
simp [one_apply]

Premises used: ArithmeticFunction.one_apply
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
Goal:
R : Type u_1
inst : AddMonoid R
f g : ArithmeticFunction R
⊢ HAdd.hAdd (α := (a : ℕ) → R) (β := (a : ℕ) → R) (⇑f : (a : ℕ) → R) (⇑g : (a : ℕ) → R) 0 = 0

Tactic:
simp
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
Goal:
R : Type u_1
inst : AddMonoid R
f g : ArithmeticFunction R
n : ℕ
⊢ (f + g : (a : ℕ) → R) n = HAdd.hAdd (α := R) (β := R) ((f : (a : ℕ) → R) n : R) ((g : (a : ℕ) → R) n : R)

Tactic:
rfl

Premises used: rfl
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
Goal:
R : Type u_1
inst : AddMonoidWithOne R
n : ℕ
⊢ Eq.{u_1 + 1} (α := R) ((fun (x : ℕ) ↦ (if x = 1 then (↑n : R) else 0 : R)) 0) 0

Tactic:
simp
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
Goal:
R : Type u_1
inst : AddMonoidWithOne R
h :
  ∀ (x : ℕ),
    Eq (α := R)
      ((({ toFun := fun (x : ℕ) ↦ if x = 1 then (↑0 : R) else 0, map_zero' := ⋯ } : ZeroHom ℕ R) : (a : ℕ) → R) x : R)
      ((0 : (a : ℕ) → R) x : 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)

Tactic:
ext
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
Goal:
R : Type u_1
inst : AddMonoidWithOne R
x : ℕ
⊢ Eq (α := R)
    ((({ toFun := fun (x : ℕ) ↦ if x = 1 then (↑0 : R) else 0, map_zero' := ⋯ } : ZeroHom ℕ R) : (a : ℕ) → R) x : R)
    ((0 : (a : ℕ) → R) x : R)

Tactic:
simp
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
Goal:
R : Type u_1
inst : AddMonoidWithOne R
n : ℕ
h :
  ∀ (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)
⊢ { 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

Tactic:
ext x
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
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)
neg :
  ∀ (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)

Tactic:
by_cases h : x = 1
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
Goal:
R : Type u_1
inst : AddMonoidWithOne R
n x : ℕ
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)

Tactic:
simp [h]
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
Goal:
R : Type u_1
inst : AddMonoidWithOne R
n x : ℕ
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)

Tactic:
simp [h]
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
Goal:
R : Type u_1
inst : NegZeroClass R
f : ArithmeticFunction R
⊢ Eq (α := R) ((-(⇑f : (a : ℕ) → R)) 0 : R) (0 : R)

Tactic:
simp
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
Goal:
R : Type u_1
inst : NegZeroClass R
f : ArithmeticFunction R
n : ℕ
⊢ Eq (α := R) ((-f : (a : ℕ) → R) n : R) (-(f : (a : ℕ) → R) n : R)

Tactic:
rfl
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
Goal:
R : Type u_1
M : Type u_2
inst : Zero R
inst_1 : AddCommMonoid M
inst_2 : SMul R M
f : ArithmeticFunction R
g : ArithmeticFunction M
⊢ Eq.{u_2 + 1} (α := M)
    ((fun (n : ℕ) ↦
        (∑ x ∈ n.divisorsAntidiagonal,
            HSMul.hSMul (α := R) (β := M) ((f : (a : ℕ) → R) x.1 : R) ((g : (a : ℕ) → M) x.2 : M) :
          M))
      0)
    0

Tactic:
simp
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
Goal:
R : Type u_1
M : Type u_2
inst : Zero R
inst_1 : AddCommMonoid M
inst_2 : SMul R M
f : ArithmeticFunction R
g : ArithmeticFunction M
n : ℕ
⊢ Eq (α := M) ((f • g : (a : ℕ) → M) n : M)
    (∑ x ∈ n.divisorsAntidiagonal,
        HSMul.hSMul (α := R) (β := M) ((f : (a : ℕ) → R) x.1 : R) ((g : (a : ℕ) → M) x.2 : M) :
      M)

Tactic:
rfl

Premises used: Multiset.sum, Multiset.foldr, Finset.sum, rfl
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
Goal:
R : Type u_1
inst : Semiring R
f g : ArithmeticFunction R
n : ℕ
⊢ Eq (α := R) ((f * g : (a : ℕ) → R) n : R)
    (∑ x ∈ n.divisorsAntidiagonal, HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) x.1 : R) ((g : (a : ℕ) → R) x.2 : R) :
      R)

Tactic:
rfl

Premises used: Multiset.sum, Multiset.foldr, Finset.sum, rfl
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
Goal:
R : Type u_1
inst : Semiring R
f g : ArithmeticFunction R
⊢ (f * g : (a : ℕ) → R) 1 = HMul.hMul (α := R) (β := R) ((f : (a : ℕ) → R) 1 : R) ((g : (a : ℕ) → R) 1 : R)

Tactic:
simp
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
Goal:
R : Type u_1
inst : Semiring R
f g : ArithmeticFunction ℕ
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)
⊢ (↑(f * g) : ArithmeticFunction R) =
    HMul.hMul (α := ArithmeticFunction R) (↑f : ArithmeticFunction R) (↑g : ArithmeticFunction R)

Tactic:
ext n
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
Goal:
R : Type u_1
inst : Semiring R
f g : ArithmeticFunction ℕ
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:
simp
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
Goal:
R : Type u_1
inst : Ring R
f g : ArithmeticFunction ℤ
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)
⊢ (↑(f * g) : ArithmeticFunction R) =
    HMul.hMul (α := ArithmeticFunction R) (↑f : ArithmeticFunction R) (↑g : ArithmeticFunction R)

Tactic:
ext n
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
Goal:
R : Type u_1
inst : Ring R
f g : ArithmeticFunction ℤ
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:
simp
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
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
h₁ : ∀ (n : ℕ), Eq (α := M) (((f * g) • h : (a : ℕ) → M) n : M) ((f • g • h : (a : ℕ) → M) n : M)
⊢ (f * g) • h = f • g • h

Tactic:
ext n
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
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 :
  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)))
⊢ Eq (α := M) (((f * g) • h : (a : ℕ) → M) n : M) ((f • g • h : (a : ℕ) → M) n : M)

Tactic:
simp only [mul_apply, smul_apply, sum_smul, mul_smul, smul_sum, sum_sigma']

Premises used: Finset.sum_smul, ArithmeticFunction.smul_apply, Finset.sum_sigma', Finset.smul_sum, ArithmeticFunction.mul_apply, SemigroupAction.mul_smul
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
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
⊢ ∀ (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:
aesop (add simp mul_assoc)

Premises used: mul_assoc
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
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
⊢ ∀ (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:
aesop (add simp mul_assoc)

Premises used: mul_assoc
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
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
⊢ ∀ (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:
aesop (add simp mul_assoc)

Premises used: mul_assoc
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
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
⊢ ∀ (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 →
      (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) (i * k, l) (i, k) with
        | ⟨(_i, j), (k, l)⟩ => Sigma.mk (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ) (k, l * j) (l, j)) =
        a

Tactic:
aesop (add simp mul_assoc)

Premises used: mul_assoc
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
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
⊢ ∀ (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 →
      HSMul.hSMul (α := R) ((f : (a : ℕ) → R) a.snd (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ).1 : R)
          (HSMul.hSMul (α := R) (β := M) ((g : (a : ℕ) → R) a.snd (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ).2 : R)
            ((h : (a : ℕ) → M) a.fst (β := fun (i : ℕ × ℕ) ↦ ℕ × ℕ).2 : M)) =
        HSMul.hSMul (α := R)
          ((f : (a : ℕ) → R)
              (Sigma.fst (β := fun (i : 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))).1 :
            R)
          (HSMul.hSMul (α := R) (β := M)
            ((g : (a : ℕ) → R)
                (Sigma.snd (β := fun (i : 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))).1 :
              R)
            ((h : (a : ℕ) → M)
                (Sigma.snd (β := fun (i : 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))).2 :
              M))

Tactic:
aesop (add simp mul_assoc)

Premises used: mul_assoc
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
Goal:
R : Type u_1
M : Type u_2
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
b : ArithmeticFunction M
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)
⊢ HSMul.hSMul.{u_1, u_2, u_2} (α := ArithmeticFunction R) 1 b = b

Tactic:
ext x
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
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
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)
⊢ 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:
rw [smul_apply]

Premises used: ArithmeticFunction.smul_apply
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
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)
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)
⊢ 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
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
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)

Tactic:
simp [x0]
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
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
⊢ {(1, x)} ⊆ x.divisorsAntidiagonal

Tactic:
simp [x0]
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
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
⊢ {(1, x)} ⊆ x.divisorsAntidiagonal

Tactic:
have h : {(1, x)} ⊆ divisorsAntidiagonal x := by simp [x0]
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
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)
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
⊢ 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 [← sum_subset h]

Premises used: Finset.sum_subset
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
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 ∈ {(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:
simp
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
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)}
con : y.1 = 1
⊢ False

Tactic:
simp_all [Prod.ext_iff]

Premises used: Prod.ext_iff
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
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)}
⊢ y.1 ≠ 1

Tactic:
have y1ne : y.fst ≠ 1 := fun con => by simp_all [Prod.ext_iff]
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
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)}
y1ne : y.1 ≠ 1
⊢ 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

Tactic:
simp [y1ne]
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
Goal:
R : Type u_1
inst : Semiring R
f : ArithmeticFunction R
h : ∀ (x : ℕ), Eq (α := R) ((f * 1 : (a : ℕ) → R) x : R) ((f : (a : ℕ) → R) x : R)
⊢ f * 1 = f

Tactic:
ext x
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
Goal:
R : Type u_1
inst : Semiring R
f : ArithmeticFunction R
x : ℕ
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)
⊢ Eq (α := R) ((f * 1 : (a : ℕ) → R) x : R) ((f : (a : ℕ) → R) x : R)

Tactic:
rw [mul_apply]

Premises used: ArithmeticFunction.mul_apply
-- Module: Mathlib.NumberTheory.ArithmeticFunction.Defs
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)
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)
⊢ 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