l3lab/ntp-mathlib huggingface dataset:
theorem mul_smul' (f g : ArithmeticFunction R) (h : ArithmeticFunction M) :
    (f * g) • h = f • g • h := by
  ext n
  simp only [mul_apply, smul_apply, sum_smul, mul_smul, smul_sum, Finset.sum_sigma']
  apply Finset.sum_bij
  pick_goal 5
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ _H
    exact ⟨(k, l * j), (l, j)⟩
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ H
    simp only [Finset.mem_sigma, mem_divisorsAntidiagonal] at H ⊢
    rcases H with ⟨⟨rfl, n0⟩, rfl, i0⟩
    refine' ⟨⟨(mul_assoc _ _ _).symm, n0⟩, trivial, _⟩
    rw [mul_ne_zero_iff] at *
    exact ⟨i0.2, n0.2⟩
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ _H
    simp only [mul_assoc]
  · rintro ⟨⟨a, b⟩, ⟨c, d⟩⟩ ⟨⟨i, j⟩, ⟨k, l⟩⟩ H₁ H₂
    simp only [Finset.mem_sigma, mem_divisorsAntidiagonal, and_imp, Prod.mk.inj_iff, add_comm,
      heq_iff_eq] at H₁ H₂ ⊢
    simp only [Sigma.mk.inj_iff, Prod.mk.injEq, heq_eq_eq, and_imp] -- porting note: added
    rintro h h2 rfl rfl
    subst h -- porting note: added.  The `rintro h ...` above was `rintro rfl ...`
    exact ⟨⟨Eq.trans H₁.2.1.symm H₂.2.1, rfl⟩, rfl, rfl⟩
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ H
    refine' ⟨⟨(i * k, l), (i, k)⟩, _, _⟩
    · simp only [Finset.mem_sigma, mem_divisorsAntidiagonal] at H ⊢
      rcases H with ⟨⟨rfl, n0⟩, rfl, j0⟩
      refine' ⟨⟨mul_assoc _ _ _, n0⟩, trivial, _⟩
      rw [mul_ne_zero_iff] at *
      

Proof state:
case h.i_surj.mk.mk.mk.refine'_1.intro.intro.intro
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
f g : ArithmeticFunction R
h : ArithmeticFunction M
i k l : ℕ
n0 : i ≠ 0 ∧ k * l ≠ 0
j0 : k ≠ 0 ∧ l ≠ 0
⊢ i ≠ 0 ∧ k ≠ 0

Next tactic:
exact ⟨n0.1, j0.1⟩
theorem mul_smul' (f g : ArithmeticFunction R) (h : ArithmeticFunction M) :
    (f * g) • h = f • g • h := by
  ext n
  simp only [mul_apply, smul_apply, sum_smul, mul_smul, smul_sum, Finset.sum_sigma']
  apply Finset.sum_bij
  pick_goal 5
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ _H
    exact ⟨(k, l * j), (l, j)⟩
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ H
    simp only [Finset.mem_sigma, mem_divisorsAntidiagonal] at H ⊢
    rcases H with ⟨⟨rfl, n0⟩, rfl, i0⟩
    refine' ⟨⟨(mul_assoc _ _ _).symm, n0⟩, trivial, _⟩
    rw [mul_ne_zero_iff] at *
    exact ⟨i0.2, n0.2⟩
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ _H
    simp only [mul_assoc]
  · rintro ⟨⟨a, b⟩, ⟨c, d⟩⟩ ⟨⟨i, j⟩, ⟨k, l⟩⟩ H₁ H₂
    simp only [Finset.mem_sigma, mem_divisorsAntidiagonal, and_imp, Prod.mk.inj_iff, add_comm,
      heq_iff_eq] at H₁ H₂ ⊢
    simp only [Sigma.mk.inj_iff, Prod.mk.injEq, heq_eq_eq, and_imp] -- porting note: added
    rintro h h2 rfl rfl
    subst h -- porting note: added.  The `rintro h ...` above was `rintro rfl ...`
    exact ⟨⟨Eq.trans H₁.2.1.symm H₂.2.1, rfl⟩, rfl, rfl⟩
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ H
    refine' ⟨⟨(i * k, l), (i, k)⟩, _, _⟩
    · simp only [Finset.mem_sigma, mem_divisorsAntidiagonal] at H ⊢
      rcases H with ⟨⟨rfl, n0⟩, rfl, j0⟩
      refine' ⟨⟨mul_assoc _ _ _, n0⟩, trivial, _⟩
      rw [mul_ne_zero_iff] at *
      exact ⟨n0.1, j0.1⟩
    · 

Proof state:
case h.i_surj.mk.mk.mk.refine'_2
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
f g : ArithmeticFunction R
h : ArithmeticFunction M
n i j k l : ℕ
H : { fst := (i, j), snd := (k, l) } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.2
⊢ { fst := (i, j), snd := (k, l) } =
    Sigma.casesOn (motive := fun x =>
      (x ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.1) → (_ : ℕ × ℕ) × ℕ × ℕ)
      { fst := (i * k, l), snd := (i, k) }
      (fun fst snd =>
        Prod.casesOn (motive := fun x =>
          ({ fst := x, snd := snd } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.1) →
            (_ : ℕ × ℕ) × ℕ × ℕ)
          fst fun i j =>
          Prod.casesOn (motive := fun x =>
            ({ fst := (i, j), snd := x } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.1) →
              (_ : ℕ × ℕ) × ℕ × ℕ)
            snd fun k l _H => { fst := (k, l * j), snd := (l, j) })
      (_ :
        { fst := (i * k, l), snd := (i, k) } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.1)

Next tactic:
simp only [true_and_iff, mem_divisorsAntidiagonal, and_true_iff, Prod.mk.inj_iff,
        eq_self_iff_true, Ne.def, mem_sigma, heq_iff_eq] at H ⊢
theorem mul_smul' (f g : ArithmeticFunction R) (h : ArithmeticFunction M) :
    (f * g) • h = f • g • h := by
  ext n
  simp only [mul_apply, smul_apply, sum_smul, mul_smul, smul_sum, Finset.sum_sigma']
  apply Finset.sum_bij
  pick_goal 5
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ _H
    exact ⟨(k, l * j), (l, j)⟩
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ H
    simp only [Finset.mem_sigma, mem_divisorsAntidiagonal] at H ⊢
    rcases H with ⟨⟨rfl, n0⟩, rfl, i0⟩
    refine' ⟨⟨(mul_assoc _ _ _).symm, n0⟩, trivial, _⟩
    rw [mul_ne_zero_iff] at *
    exact ⟨i0.2, n0.2⟩
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ _H
    simp only [mul_assoc]
  · rintro ⟨⟨a, b⟩, ⟨c, d⟩⟩ ⟨⟨i, j⟩, ⟨k, l⟩⟩ H₁ H₂
    simp only [Finset.mem_sigma, mem_divisorsAntidiagonal, and_imp, Prod.mk.inj_iff, add_comm,
      heq_iff_eq] at H₁ H₂ ⊢
    simp only [Sigma.mk.inj_iff, Prod.mk.injEq, heq_eq_eq, and_imp] -- porting note: added
    rintro h h2 rfl rfl
    subst h -- porting note: added.  The `rintro h ...` above was `rintro rfl ...`
    exact ⟨⟨Eq.trans H₁.2.1.symm H₂.2.1, rfl⟩, rfl, rfl⟩
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ H
    refine' ⟨⟨(i * k, l), (i, k)⟩, _, _⟩
    · simp only [Finset.mem_sigma, mem_divisorsAntidiagonal] at H ⊢
      rcases H with ⟨⟨rfl, n0⟩, rfl, j0⟩
      refine' ⟨⟨mul_assoc _ _ _, n0⟩, trivial, _⟩
      rw [mul_ne_zero_iff] at *
      exact ⟨n0.1, j0.1⟩
    · simp only [true_and_iff, mem_divisorsAntidiagonal, and_true_iff, Prod.mk.inj_iff,
        eq_self_iff_true, Ne.def, mem_sigma, heq_iff_eq] at H ⊢
      

Proof state:
case h.i_surj.mk.mk.mk.refine'_2
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
f g : ArithmeticFunction R
h : ArithmeticFunction M
n i j k l : ℕ
H : (i * j = n ∧ ¬n = 0) ∧ k * l = j ∧ ¬j = 0
⊢ { fst := (i, j), snd := (k, l) } = { fst := (i, k * l), snd := (k, l) }

Next tactic:
rw [H.2.1]
theorem one_smul' (b : ArithmeticFunction M) : (1 : ArithmeticFunction R) • b = b := by
  

Proof state:
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
b : ArithmeticFunction M
⊢ 1 • b = b

Next tactic:
ext x
theorem one_smul' (b : ArithmeticFunction M) : (1 : ArithmeticFunction R) • b = b := by
  ext x
  

Proof state:
case h
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
b : ArithmeticFunction M
x : ℕ
⊢ (1 • b) x = b x

Next tactic:
rw [smul_apply]
theorem one_smul' (b : ArithmeticFunction M) : (1 : ArithmeticFunction R) • b = b := by
  ext x
  rw [smul_apply]
  

Proof state:
case h
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
b : ArithmeticFunction M
x : ℕ
⊢ ∑ x in divisorsAntidiagonal x, 1 x.1 • b x.2 = b x

Next tactic:
by_cases x0 : x = 0
theorem one_smul' (b : ArithmeticFunction M) : (1 : ArithmeticFunction R) • b = b := by
  ext x
  rw [smul_apply]
  by_cases x0 : x = 0
  · 

Proof state:
case pos
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
b : ArithmeticFunction M
x : ℕ
x0 : x = 0
⊢ ∑ x in divisorsAntidiagonal x, 1 x.1 • b x.2 = b x

Next tactic:
simp [x0]
theorem one_smul' (b : ArithmeticFunction M) : (1 : ArithmeticFunction R) • b = b := by
  ext x
  rw [smul_apply]
  by_cases x0 : x = 0
  · simp [x0]
  

Proof state:
case neg
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
b : ArithmeticFunction M
x : ℕ
x0 : ¬x = 0
⊢ ∑ x in divisorsAntidiagonal x, 1 x.1 • b x.2 = b x

Next tactic:
have h : {(1, x)} ⊆ divisorsAntidiagonal x := by simp [x0]
theorem one_smul' (b : ArithmeticFunction M) : (1 : ArithmeticFunction R) • b = b := by
  ext x
  rw [smul_apply]
  by_cases x0 : x = 0
  · simp [x0]
  have h : {(1, x)} ⊆ divisorsAntidiagonal x := by 

Proof state:
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
b : ArithmeticFunction M
x : ℕ
x0 : ¬x = 0
⊢ {(1, x)} ⊆ divisorsAntidiagonal x

Next tactic:
simp [x0]
theorem one_smul' (b : ArithmeticFunction M) : (1 : ArithmeticFunction R) • b = b := by
  ext x
  rw [smul_apply]
  by_cases x0 : x = 0
  · simp [x0]
  have h : {(1, x)} ⊆ divisorsAntidiagonal x := by simp [x0]
  

Proof state:
case neg
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
b : ArithmeticFunction M
x : ℕ
x0 : ¬x = 0
h : {(1, x)} ⊆ divisorsAntidiagonal x
⊢ ∑ x in divisorsAntidiagonal x, 1 x.1 • b x.2 = b x

Next tactic:
rw [← sum_subset h]
theorem one_smul' (b : ArithmeticFunction M) : (1 : ArithmeticFunction R) • b = b := by
  ext x
  rw [smul_apply]
  by_cases x0 : x = 0
  · simp [x0]
  have h : {(1, x)} ⊆ divisorsAntidiagonal x := by simp [x0]
  rw [← sum_subset h]
  · 

Proof state:
case neg
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
b : ArithmeticFunction M
x : ℕ
x0 : ¬x = 0
h : {(1, x)} ⊆ divisorsAntidiagonal x
⊢ ∑ x in {(1, x)}, 1 x.1 • b x.2 = b x

Next tactic:
simp
theorem one_smul' (b : ArithmeticFunction M) : (1 : ArithmeticFunction R) • b = b := by
  ext x
  rw [smul_apply]
  by_cases x0 : x = 0
  · simp [x0]
  have h : {(1, x)} ⊆ divisorsAntidiagonal x := by simp [x0]
  rw [← sum_subset h]
  · simp
  

Proof state:
case neg
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
b : ArithmeticFunction M
x : ℕ
x0 : ¬x = 0
h : {(1, x)} ⊆ divisorsAntidiagonal x
⊢ ∀ x_1 ∈ divisorsAntidiagonal x, x_1 ∉ {(1, x)} → 1 x_1.1 • b x_1.2 = 0

Next tactic:
intro y ymem ynmem
theorem one_smul' (b : ArithmeticFunction M) : (1 : ArithmeticFunction R) • b = b := by
  ext x
  rw [smul_apply]
  by_cases x0 : x = 0
  · simp [x0]
  have h : {(1, x)} ⊆ divisorsAntidiagonal x := by simp [x0]
  rw [← sum_subset h]
  · simp
  intro y ymem ynmem
  

Proof state:
case neg
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
b : ArithmeticFunction M
x : ℕ
x0 : ¬x = 0
h : {(1, x)} ⊆ divisorsAntidiagonal x
y : ℕ × ℕ
ymem : y ∈ divisorsAntidiagonal x
ynmem : y ∉ {(1, x)}
⊢ 1 y.1 • b y.2 = 0

Next tactic:
have y1ne : y.fst ≠ 1 := by
    intro con
    simp only [Con, mem_divisorsAntidiagonal, one_mul, Ne.def] at ymem
    simp only [mem_singleton, Prod.ext_iff] at ynmem
    -- porting note: `tauto` worked from here.
    cases y
    subst con
    simp only [true_and, one_mul, x0, not_false_eq_true, and_true] at ynmem ymem
    tauto
theorem one_smul' (b : ArithmeticFunction M) : (1 : ArithmeticFunction R) • b = b := by
  ext x
  rw [smul_apply]
  by_cases x0 : x = 0
  · simp [x0]
  have h : {(1, x)} ⊆ divisorsAntidiagonal x := by simp [x0]
  rw [← sum_subset h]
  · simp
  intro y ymem ynmem
  have y1ne : y.fst ≠ 1 := by
    

Proof state:
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
b : ArithmeticFunction M
x : ℕ
x0 : ¬x = 0
h : {(1, x)} ⊆ divisorsAntidiagonal x
y : ℕ × ℕ
ymem : y ∈ divisorsAntidiagonal x
ynmem : y ∉ {(1, x)}
⊢ y.1 ≠ 1

Next tactic:
intro con
theorem one_smul' (b : ArithmeticFunction M) : (1 : ArithmeticFunction R) • b = b := by
  ext x
  rw [smul_apply]
  by_cases x0 : x = 0
  · simp [x0]
  have h : {(1, x)} ⊆ divisorsAntidiagonal x := by simp [x0]
  rw [← sum_subset h]
  · simp
  intro y ymem ynmem
  have y1ne : y.fst ≠ 1 := by
    intro con
    

Proof state:
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
b : ArithmeticFunction M
x : ℕ
x0 : ¬x = 0
h : {(1, x)} ⊆ divisorsAntidiagonal x
y : ℕ × ℕ
ymem : y ∈ divisorsAntidiagonal x
ynmem : y ∉ {(1, x)}
con : y.1 = 1
⊢ False

Next tactic:
simp only [Con, mem_divisorsAntidiagonal, one_mul, Ne.def] at ymem
theorem one_smul' (b : ArithmeticFunction M) : (1 : ArithmeticFunction R) • b = b := by
  ext x
  rw [smul_apply]
  by_cases x0 : x = 0
  · simp [x0]
  have h : {(1, x)} ⊆ divisorsAntidiagonal x := by simp [x0]
  rw [← sum_subset h]
  · simp
  intro y ymem ynmem
  have y1ne : y.fst ≠ 1 := by
    intro con
    simp only [Con, mem_divisorsAntidiagonal, one_mul, Ne.def] at ymem
    

Proof state:
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
b : ArithmeticFunction M
x : ℕ
x0 : ¬x = 0
h : {(1, x)} ⊆ divisorsAntidiagonal x
y : ℕ × ℕ
ynmem : y ∉ {(1, x)}
con : y.1 = 1
ymem : y.1 * y.2 = x ∧ ¬x = 0
⊢ False

Next tactic:
simp only [mem_singleton, Prod.ext_iff] at ynmem
theorem one_smul' (b : ArithmeticFunction M) : (1 : ArithmeticFunction R) • b = b := by
  ext x
  rw [smul_apply]
  by_cases x0 : x = 0
  · simp [x0]
  have h : {(1, x)} ⊆ divisorsAntidiagonal x := by simp [x0]
  rw [← sum_subset h]
  · simp
  intro y ymem ynmem
  have y1ne : y.fst ≠ 1 := by
    intro con
    simp only [Con, mem_divisorsAntidiagonal, one_mul, Ne.def] at ymem
    simp only [mem_singleton, Prod.ext_iff] at ynmem
    -- porting note: `tauto` worked from here.
    

Proof state:
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
b : ArithmeticFunction M
x : ℕ
x0 : ¬x = 0
h : {(1, x)} ⊆ divisorsAntidiagonal x
y : ℕ × ℕ
con : y.1 = 1
ymem : y.1 * y.2 = x ∧ ¬x = 0
ynmem : ¬(y.1 = 1 ∧ y.2 = x)
⊢ False

Next tactic:
cases y
theorem one_smul' (b : ArithmeticFunction M) : (1 : ArithmeticFunction R) • b = b := by
  ext x
  rw [smul_apply]
  by_cases x0 : x = 0
  · simp [x0]
  have h : {(1, x)} ⊆ divisorsAntidiagonal x := by simp [x0]
  rw [← sum_subset h]
  · simp
  intro y ymem ynmem
  have y1ne : y.fst ≠ 1 := by
    intro con
    simp only [Con, mem_divisorsAntidiagonal, one_mul, Ne.def] at ymem
    simp only [mem_singleton, Prod.ext_iff] at ynmem
    -- porting note: `tauto` worked from here.
    cases y
    

Proof state:
case mk
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
b : ArithmeticFunction M
x : ℕ
x0 : ¬x = 0
h : {(1, x)} ⊆ divisorsAntidiagonal x
fst✝ snd✝ : ℕ
con : (fst✝, snd✝).1 = 1
ymem : (fst✝, snd✝).1 * (fst✝, snd✝).2 = x ∧ ¬x = 0
ynmem : ¬((fst✝, snd✝).1 = 1 ∧ (fst✝, snd✝).2 = x)
⊢ False

Next tactic:
subst con
theorem one_smul' (b : ArithmeticFunction M) : (1 : ArithmeticFunction R) • b = b := by
  ext x
  rw [smul_apply]
  by_cases x0 : x = 0
  · simp [x0]
  have h : {(1, x)} ⊆ divisorsAntidiagonal x := by simp [x0]
  rw [← sum_subset h]
  · simp
  intro y ymem ynmem
  have y1ne : y.fst ≠ 1 := by
    intro con
    simp only [Con, mem_divisorsAntidiagonal, one_mul, Ne.def] at ymem
    simp only [mem_singleton, Prod.ext_iff] at ynmem
    -- porting note: `tauto` worked from here.
    cases y
    subst con
    

Proof state:
case mk
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
b : ArithmeticFunction M
x : ℕ
x0 : ¬x = 0
h : {(1, x)} ⊆ divisorsAntidiagonal x
snd✝ : ℕ
ymem : (1, snd✝).1 * (1, snd✝).2 = x ∧ ¬x = 0
ynmem : ¬((1, snd✝).1 = 1 ∧ (1, snd✝).2 = x)
⊢ False

Next tactic:
simp only [true_and, one_mul, x0, not_false_eq_true, and_true] at ynmem ymem
theorem one_smul' (b : ArithmeticFunction M) : (1 : ArithmeticFunction R) • b = b := by
  ext x
  rw [smul_apply]
  by_cases x0 : x = 0
  · simp [x0]
  have h : {(1, x)} ⊆ divisorsAntidiagonal x := by simp [x0]
  rw [← sum_subset h]
  · simp
  intro y ymem ynmem
  have y1ne : y.fst ≠ 1 := by
    intro con
    simp only [Con, mem_divisorsAntidiagonal, one_mul, Ne.def] at ymem
    simp only [mem_singleton, Prod.ext_iff] at ynmem
    -- porting note: `tauto` worked from here.
    cases y
    subst con
    simp only [true_and, one_mul, x0, not_false_eq_true, and_true] at ynmem ymem
    

Proof state:
case mk
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
b : ArithmeticFunction M
x : ℕ
x0 : ¬x = 0
h : {(1, x)} ⊆ divisorsAntidiagonal x
snd✝ : ℕ
ynmem : ¬snd✝ = x
ymem : snd✝ = x
⊢ False

Next tactic:
tauto
theorem one_smul' (b : ArithmeticFunction M) : (1 : ArithmeticFunction R) • b = b := by
  ext x
  rw [smul_apply]
  by_cases x0 : x = 0
  · simp [x0]
  have h : {(1, x)} ⊆ divisorsAntidiagonal x := by simp [x0]
  rw [← sum_subset h]
  · simp
  intro y ymem ynmem
  have y1ne : y.fst ≠ 1 := by
    intro con
    simp only [Con, mem_divisorsAntidiagonal, one_mul, Ne.def] at ymem
    simp only [mem_singleton, Prod.ext_iff] at ynmem
    -- porting note: `tauto` worked from here.
    cases y
    subst con
    simp only [true_and, one_mul, x0, not_false_eq_true, and_true] at ynmem ymem
    tauto

  

Proof state:
case neg
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
b : ArithmeticFunction M
x : ℕ
x0 : ¬x = 0
h : {(1, x)} ⊆ divisorsAntidiagonal x
y : ℕ × ℕ
ymem : y ∈ divisorsAntidiagonal x
ynmem : y ∉ {(1, x)}
y1ne : y.1 ≠ 1
⊢ 1 y.1 • b y.2 = 0

Next tactic:
simp [y1ne]
instance instMonoid : Monoid (ArithmeticFunction R) :=
  { one := One.one
    mul := Mul.mul
    one_mul := one_smul'
    mul_one := fun f => by
      

Proof state:
R : Type u_1
inst✝ : Semiring R
f : ArithmeticFunction R
⊢ f * 1 = f

Next tactic:
ext x
instance instMonoid : Monoid (ArithmeticFunction R) :=
  { one := One.one
    mul := Mul.mul
    one_mul := one_smul'
    mul_one := fun f => by
      ext x
      

Proof state:
case h
R : Type u_1
inst✝ : Semiring R
f : ArithmeticFunction R
x : ℕ
⊢ (f * 1) x = f x

Next tactic:
rw [mul_apply]
instance instMonoid : Monoid (ArithmeticFunction R) :=
  { one := One.one
    mul := Mul.mul
    one_mul := one_smul'
    mul_one := fun f => by
      ext x
      rw [mul_apply]
      

Proof state:
case h
R : Type u_1
inst✝ : Semiring R
f : ArithmeticFunction R
x : ℕ
⊢ ∑ x in divisorsAntidiagonal x, f x.1 * 1 x.2 = f x

Next tactic:
by_cases x0 : x = 0
instance instMonoid : Monoid (ArithmeticFunction R) :=
  { one := One.one
    mul := Mul.mul
    one_mul := one_smul'
    mul_one := fun f => by
      ext x
      rw [mul_apply]
      by_cases x0 : x = 0
      · 

Proof state:
case pos
R : Type u_1
inst✝ : Semiring R
f : ArithmeticFunction R
x : ℕ
x0 : x = 0
⊢ ∑ x in divisorsAntidiagonal x, f x.1 * 1 x.2 = f x

Next tactic:
simp [x0]
instance instMonoid : Monoid (ArithmeticFunction R) :=
  { one := One.one
    mul := Mul.mul
    one_mul := one_smul'
    mul_one := fun f => by
      ext x
      rw [mul_apply]
      by_cases x0 : x = 0
      · simp [x0]
      

Proof state:
case neg
R : Type u_1
inst✝ : Semiring R
f : ArithmeticFunction R
x : ℕ
x0 : ¬x = 0
⊢ ∑ x in divisorsAntidiagonal x, f x.1 * 1 x.2 = f x

Next tactic:
have h : {(x, 1)} ⊆ divisorsAntidiagonal x := by simp [x0]
instance instMonoid : Monoid (ArithmeticFunction R) :=
  { one := One.one
    mul := Mul.mul
    one_mul := one_smul'
    mul_one := fun f => by
      ext x
      rw [mul_apply]
      by_cases x0 : x = 0
      · simp [x0]
      have h : {(x, 1)} ⊆ divisorsAntidiagonal x := by 

Proof state:
R : Type u_1
inst✝ : Semiring R
f : ArithmeticFunction R
x : ℕ
x0 : ¬x = 0
⊢ {(x, 1)} ⊆ divisorsAntidiagonal x

Next tactic:
simp [x0]
instance instMonoid : Monoid (ArithmeticFunction R) :=
  { one := One.one
    mul := Mul.mul
    one_mul := one_smul'
    mul_one := fun f => by
      ext x
      rw [mul_apply]
      by_cases x0 : x = 0
      · simp [x0]
      have h : {(x, 1)} ⊆ divisorsAntidiagonal x := by simp [x0]
      

Proof state:
case neg
R : Type u_1
inst✝ : Semiring R
f : ArithmeticFunction R
x : ℕ
x0 : ¬x = 0
h : {(x, 1)} ⊆ divisorsAntidiagonal x
⊢ ∑ x in divisorsAntidiagonal x, f x.1 * 1 x.2 = f x

Next tactic:
rw [← sum_subset h]
instance instMonoid : Monoid (ArithmeticFunction R) :=
  { one := One.one
    mul := Mul.mul
    one_mul := one_smul'
    mul_one := fun f => by
      ext x
      rw [mul_apply]
      by_cases x0 : x = 0
      · simp [x0]
      have h : {(x, 1)} ⊆ divisorsAntidiagonal x := by simp [x0]
      rw [← sum_subset h]
      · 

Proof state:
case neg
R : Type u_1
inst✝ : Semiring R
f : ArithmeticFunction R
x : ℕ
x0 : ¬x = 0
h : {(x, 1)} ⊆ divisorsAntidiagonal x
⊢ ∑ x in {(x, 1)}, f x.1 * 1 x.2 = f x

Next tactic:
simp
instance instMonoid : Monoid (ArithmeticFunction R) :=
  { one := One.one
    mul := Mul.mul
    one_mul := one_smul'
    mul_one := fun f => by
      ext x
      rw [mul_apply]
      by_cases x0 : x = 0
      · simp [x0]
      have h : {(x, 1)} ⊆ divisorsAntidiagonal x := by simp [x0]
      rw [← sum_subset h]
      · simp
      

Proof state:
case neg
R : Type u_1
inst✝ : Semiring R
f : ArithmeticFunction R
x : ℕ
x0 : ¬x = 0
h : {(x, 1)} ⊆ divisorsAntidiagonal x
⊢ ∀ x_1 ∈ divisorsAntidiagonal x, x_1 ∉ {(x, 1)} → f x_1.1 * 1 x_1.2 = 0

Next tactic:
intro y ymem ynmem
instance instMonoid : Monoid (ArithmeticFunction R) :=
  { one := One.one
    mul := Mul.mul
    one_mul := one_smul'
    mul_one := fun f => by
      ext x
      rw [mul_apply]
      by_cases x0 : x = 0
      · simp [x0]
      have h : {(x, 1)} ⊆ divisorsAntidiagonal x := by simp [x0]
      rw [← sum_subset h]
      · simp
      intro y ymem ynmem
      

Proof state:
case neg
R : Type u_1
inst✝ : Semiring R
f : ArithmeticFunction R
x : ℕ
x0 : ¬x = 0
h : {(x, 1)} ⊆ divisorsAntidiagonal x
y : ℕ × ℕ
ymem : y ∈ divisorsAntidiagonal x
ynmem : y ∉ {(x, 1)}
⊢ f y.1 * 1 y.2 = 0

Next tactic:
have y2ne : y.snd ≠ 1 := by
        intro con
        cases y; subst con -- porting note: added
        simp only [Con, mem_divisorsAntidiagonal, mul_one, Ne.def] at ymem
        simp only [mem_singleton, Prod.ext_iff] at ynmem
        tauto
instance instMonoid : Monoid (ArithmeticFunction R) :=
  { one := One.one
    mul := Mul.mul
    one_mul := one_smul'
    mul_one := fun f => by
      ext x
      rw [mul_apply]
      by_cases x0 : x = 0
      · simp [x0]
      have h : {(x, 1)} ⊆ divisorsAntidiagonal x := by simp [x0]
      rw [← sum_subset h]
      · simp
      intro y ymem ynmem
      have y2ne : y.snd ≠ 1 := by
        

Proof state:
R : Type u_1
inst✝ : Semiring R
f : ArithmeticFunction R
x : ℕ
x0 : ¬x = 0
h : {(x, 1)} ⊆ divisorsAntidiagonal x
y : ℕ × ℕ
ymem : y ∈ divisorsAntidiagonal x
ynmem : y ∉ {(x, 1)}
⊢ y.2 ≠ 1

Next tactic:
intro con
instance instMonoid : Monoid (ArithmeticFunction R) :=
  { one := One.one
    mul := Mul.mul
    one_mul := one_smul'
    mul_one := fun f => by
      ext x
      rw [mul_apply]
      by_cases x0 : x = 0
      · simp [x0]
      have h : {(x, 1)} ⊆ divisorsAntidiagonal x := by simp [x0]
      rw [← sum_subset h]
      · simp
      intro y ymem ynmem
      have y2ne : y.snd ≠ 1 := by
        intro con
        

Proof state:
R : Type u_1
inst✝ : Semiring R
f : ArithmeticFunction R
x : ℕ
x0 : ¬x = 0
h : {(x, 1)} ⊆ divisorsAntidiagonal x
y : ℕ × ℕ
ymem : y ∈ divisorsAntidiagonal x
ynmem : y ∉ {(x, 1)}
con : y.2 = 1
⊢ False

Next tactic:
cases y
instance instMonoid : Monoid (ArithmeticFunction R) :=
  { one := One.one
    mul := Mul.mul
    one_mul := one_smul'
    mul_one := fun f => by
      ext x
      rw [mul_apply]
      by_cases x0 : x = 0
      · simp [x0]
      have h : {(x, 1)} ⊆ divisorsAntidiagonal x := by simp [x0]
      rw [← sum_subset h]
      · simp
      intro y ymem ynmem
      have y2ne : y.snd ≠ 1 := by
        intro con
        cases y; 

Proof state:
case mk
R : Type u_1
inst✝ : Semiring R
f : ArithmeticFunction R
x : ℕ
x0 : ¬x = 0
h : {(x, 1)} ⊆ divisorsAntidiagonal x
fst✝ snd✝ : ℕ
ymem : (fst✝, snd✝) ∈ divisorsAntidiagonal x
ynmem : (fst✝, snd✝) ∉ {(x, 1)}
con : (fst✝, snd✝).2 = 1
⊢ False

Next tactic:
subst con
instance instMonoid : Monoid (ArithmeticFunction R) :=
  { one := One.one
    mul := Mul.mul
    one_mul := one_smul'
    mul_one := fun f => by
      ext x
      rw [mul_apply]
      by_cases x0 : x = 0
      · simp [x0]
      have h : {(x, 1)} ⊆ divisorsAntidiagonal x := by simp [x0]
      rw [← sum_subset h]
      · simp
      intro y ymem ynmem
      have y2ne : y.snd ≠ 1 := by
        intro con
        cases y; subst con -- porting note: added
        

Proof state:
case mk
R : Type u_1
inst✝ : Semiring R
f : ArithmeticFunction R
x : ℕ
x0 : ¬x = 0
h : {(x, 1)} ⊆ divisorsAntidiagonal x
fst✝ : ℕ
ymem : (fst✝, 1) ∈ divisorsAntidiagonal x
ynmem : (fst✝, 1) ∉ {(x, 1)}
⊢ False

Next tactic:
simp only [Con, mem_divisorsAntidiagonal, mul_one, Ne.def] at ymem