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 ⊢
    

Proof state:
case h.hi.mk.mk.mk
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 = i ∧ i ≠ 0
⊢ (k * (l * j) = n ∧ n ≠ 0) ∧ True ∧ l * j ≠ 0

Next tactic:
rcases H with ⟨⟨rfl, n0⟩, rfl, i0⟩
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⟩
    

Proof state:
case h.hi.mk.mk.mk.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
j k l : ℕ
n0 : k * l * j ≠ 0
i0 : k * l ≠ 0
⊢ (k * (l * j) = k * l * j ∧ k * l * j ≠ 0) ∧ True ∧ l * j ≠ 0

Next tactic:
refine' ⟨⟨(mul_assoc _ _ _).symm, n0⟩, trivial, _⟩
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, _⟩
    

Proof state:
case h.hi.mk.mk.mk.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
j k l : ℕ
n0 : k * l * j ≠ 0
i0 : k * l ≠ 0
⊢ l * j ≠ 0

Next tactic:
rw [mul_ne_zero_iff] at *
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 *
    

Proof state:
case h.hi.mk.mk.mk.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
j k l : ℕ
n0 : k * l ≠ 0 ∧ j ≠ 0
i0 : k ≠ 0 ∧ l ≠ 0
⊢ l ≠ 0 ∧ j ≠ 0

Next tactic:
exact ⟨i0.2, n0.2⟩
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]
  · 

Proof state:
case h.i_inj
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 : ℕ
⊢ ∀ (a₁ a₂ : (_ : ℕ × ℕ) × ℕ × ℕ) (ha₁ : a₁ ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.1)
    (ha₂ : a₂ ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.1),
    Sigma.casesOn (motive := fun x =>
          (x ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.1) → (_ : ℕ × ℕ) × ℕ × ℕ) a₁
          (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) })
          ha₁ =
        Sigma.casesOn (motive := fun x =>
          (x ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.1) → (_ : ℕ × ℕ) × ℕ × ℕ) a₂
          (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) })
          ha₂ →
      a₁ = a₂

Next tactic:
rintro ⟨⟨a, b⟩, ⟨c, d⟩⟩ ⟨⟨i, j⟩, ⟨k, l⟩⟩ H₁ H₂
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
    

Proof state:
case h.i_inj.mk.mk.mk.mk.mk.mk
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 a b c d i j k l : ℕ
H₁ : { fst := (a, b), snd := (c, d) } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.1
H₂ : { fst := (i, j), snd := (k, l) } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.1
⊢ Sigma.casesOn (motive := fun x =>
        (x ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.1) → (_ : ℕ × ℕ) × ℕ × ℕ)
        { fst := (a, b), snd := (c, d) }
        (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) })
        H₁ =
      Sigma.casesOn (motive := fun x =>
        (x ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.1) → (_ : ℕ × ℕ) × ℕ × ℕ)
        { fst := (i, j), snd := (k, l) }
        (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) })
        H₂ →
    { fst := (a, b), snd := (c, d) } = { fst := (i, j), snd := (k, l) }

Next tactic:
simp only [Finset.mem_sigma, mem_divisorsAntidiagonal, and_imp, Prod.mk.inj_iff, add_comm,
      heq_iff_eq] at H₁ H₂ ⊢
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 ⊢
    

Proof state:
case h.i_inj.mk.mk.mk.mk.mk.mk
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 a b c d i j k l : ℕ
H₁ : (a * b = n ∧ n ≠ 0) ∧ c * d = a ∧ a ≠ 0
H₂ : (i * j = n ∧ n ≠ 0) ∧ k * l = i ∧ i ≠ 0
⊢ { fst := (c, d * b), snd := (d, b) } = { fst := (k, l * j), snd := (l, j) } →
    { fst := (a, b), snd := (c, d) } = { fst := (i, j), snd := (k, l) }

Next tactic:
simp only [Sigma.mk.inj_iff, Prod.mk.injEq, heq_eq_eq, and_imp]
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
    

Proof state:
case h.i_inj.mk.mk.mk.mk.mk.mk
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 a b c d i j k l : ℕ
H₁ : (a * b = n ∧ n ≠ 0) ∧ c * d = a ∧ a ≠ 0
H₂ : (i * j = n ∧ n ≠ 0) ∧ k * l = i ∧ i ≠ 0
⊢ c = k → d * b = l * j → d = l → b = j → (a = i ∧ b = j) ∧ c = k ∧ d = l

Next tactic:
rintro h h2 rfl rfl
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
    

Proof state:
case h.i_inj.mk.mk.mk.mk.mk.mk
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 a b c d i k : ℕ
H₁ : (a * b = n ∧ n ≠ 0) ∧ c * d = a ∧ a ≠ 0
h : c = k
H₂ : (i * b = n ∧ n ≠ 0) ∧ k * d = i ∧ i ≠ 0
h2 : d * b = d * b
⊢ (a = i ∧ b = b) ∧ c = k ∧ d = d

Next tactic:
subst 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 ...`
    

Proof state:
case h.i_inj.mk.mk.mk.mk.mk.mk
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 a b c d i : ℕ
H₁ : (a * b = n ∧ n ≠ 0) ∧ c * d = a ∧ a ≠ 0
h2 : d * b = d * b
H₂ : (i * b = n ∧ n ≠ 0) ∧ c * d = i ∧ i ≠ 0
⊢ (a = i ∧ b = b) ∧ c = c ∧ d = d

Next tactic:
exact ⟨⟨Eq.trans H₁.2.1.symm H₂.2.1, rfl⟩, rfl, rfl⟩
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⟩
  · 

Proof state:
case h.i_surj
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 : ℕ
⊢ ∀ b ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.2,
    ∃ a,
      ∃ (ha : a ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.1),
        b =
          Sigma.casesOn (motive := fun x =>
            (x ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.1) → (_ : ℕ × ℕ) × ℕ × ℕ) a
            (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) })
            ha

Next tactic:
rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩ 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
    

Proof state:
case h.i_surj.mk.mk.mk
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
⊢ ∃ a,
    ∃ (ha : a ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.1),
      { 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) a
          (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) })
          ha

Next tactic:
refine' ⟨⟨(i * k, l), (i, k)⟩, _, _⟩
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)⟩, _, _⟩
    · 

Proof state:
case h.i_surj.mk.mk.mk.refine'_1
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 * k, l), snd := (i, k) } ∈ Finset.sigma (divisorsAntidiagonal n) fun a => divisorsAntidiagonal a.1

Next tactic:
simp only [Finset.mem_sigma, mem_divisorsAntidiagonal] 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 ⊢
      

Proof state:
case h.i_surj.mk.mk.mk.refine'_1
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
⊢ (i * k * l = n ∧ n ≠ 0) ∧ True ∧ i * k ≠ 0

Next tactic:
rcases H with ⟨⟨rfl, n0⟩, rfl, j0⟩
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⟩
      

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 * (k * l) ≠ 0
j0 : k * l ≠ 0
⊢ (i * k * l = i * (k * l) ∧ i * (k * l) ≠ 0) ∧ True ∧ i * k ≠ 0

Next tactic:
refine' ⟨⟨mul_assoc _ _ _, n0⟩, trivial, _⟩
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, _⟩
      

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 * (k * l) ≠ 0
j0 : k * l ≠ 0
⊢ i * k ≠ 0

Next tactic:
rw [mul_ne_zero_iff] at *