l3lab/ntp-mathlib huggingface dataset:
theorem coeff_monomial_mul (a : R) :
    coeff R m (monomial R n a * φ) = if n ≤ m then a * coeff R (m - n) φ else 0 := by
  classical
  have :
    ∀ p ∈ antidiagonal m,
      coeff R (p : (σ →₀ ℕ) × (σ →₀ ℕ)).1 (monomial R n a) * coeff R p.2 φ ≠ 0 → p.1 = n :=
    fun p _ hp => eq_of_coeff_monomial_ne_zero (left_ne_zero_of_mul hp)
  rw [coeff_mul, ← Finset.sum_filter_of_ne this, Finset.filter_fst_eq_antidiagonal _ n,
    Finset.sum_ite_index]
  

Proof state:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
φ ψ : MvPowerSeries σ R
a : R
this : ∀ p ∈ antidiagonal m, (coeff R p.1) ((monomial R n) a) * (coeff R p.2) φ ≠ 0 → p.1 = n
⊢ (if n ≤ m then ∑ x in {(n, m - n)}, (coeff R x.1) ((monomial R n) a) * (coeff R x.2) φ
    else ∑ x in ∅, (coeff R x.1) ((monomial R n) a) * (coeff R x.2) φ) =
    if n ≤ m then a * (coeff R (m - n)) φ else 0

Next tactic:
simp only [Finset.sum_singleton, coeff_monomial_same, Finset.sum_empty]
theorem coeff_mul_monomial (a : R) :
    coeff R m (φ * monomial R n a) = if n ≤ m then coeff R (m - n) φ * a else 0 := by
  

Proof state:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
φ ψ : MvPowerSeries σ R
a : R
⊢ (coeff R m) (φ * (monomial R n) a) = if n ≤ m then (coeff R (m - n)) φ * a else 0

Next tactic:
classical
  have :
    ∀ p ∈ antidiagonal m,
      coeff R (p : (σ →₀ ℕ) × (σ →₀ ℕ)).1 φ * coeff R p.2 (monomial R n a) ≠ 0 → p.2 = n :=
    fun p _ hp => eq_of_coeff_monomial_ne_zero (right_ne_zero_of_mul hp)
  rw [coeff_mul, ← Finset.sum_filter_of_ne this, Finset.filter_snd_eq_antidiagonal _ n,
    Finset.sum_ite_index]
  simp only [Finset.sum_singleton, coeff_monomial_same, Finset.sum_empty]
theorem coeff_mul_monomial (a : R) :
    coeff R m (φ * monomial R n a) = if n ≤ m then coeff R (m - n) φ * a else 0 := by
  classical
  

Proof state:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
φ ψ : MvPowerSeries σ R
a : R
⊢ (coeff R m) (φ * (monomial R n) a) = if n ≤ m then (coeff R (m - n)) φ * a else 0

Next tactic:
have :
    ∀ p ∈ antidiagonal m,
      coeff R (p : (σ →₀ ℕ) × (σ →₀ ℕ)).1 φ * coeff R p.2 (monomial R n a) ≠ 0 → p.2 = n :=
    fun p _ hp => eq_of_coeff_monomial_ne_zero (right_ne_zero_of_mul hp)
theorem coeff_mul_monomial (a : R) :
    coeff R m (φ * monomial R n a) = if n ≤ m then coeff R (m - n) φ * a else 0 := by
  classical
  have :
    ∀ p ∈ antidiagonal m,
      coeff R (p : (σ →₀ ℕ) × (σ →₀ ℕ)).1 φ * coeff R p.2 (monomial R n a) ≠ 0 → p.2 = n :=
    fun p _ hp => eq_of_coeff_monomial_ne_zero (right_ne_zero_of_mul hp)
  

Proof state:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
φ ψ : MvPowerSeries σ R
a : R
this : ∀ p ∈ antidiagonal m, (coeff R p.1) φ * (coeff R p.2) ((monomial R n) a) ≠ 0 → p.2 = n
⊢ (coeff R m) (φ * (monomial R n) a) = if n ≤ m then (coeff R (m - n)) φ * a else 0

Next tactic:
rw [coeff_mul, ← Finset.sum_filter_of_ne this, Finset.filter_snd_eq_antidiagonal _ n,
    Finset.sum_ite_index]
theorem coeff_mul_monomial (a : R) :
    coeff R m (φ * monomial R n a) = if n ≤ m then coeff R (m - n) φ * a else 0 := by
  classical
  have :
    ∀ p ∈ antidiagonal m,
      coeff R (p : (σ →₀ ℕ) × (σ →₀ ℕ)).1 φ * coeff R p.2 (monomial R n a) ≠ 0 → p.2 = n :=
    fun p _ hp => eq_of_coeff_monomial_ne_zero (right_ne_zero_of_mul hp)
  rw [coeff_mul, ← Finset.sum_filter_of_ne this, Finset.filter_snd_eq_antidiagonal _ n,
    Finset.sum_ite_index]
  

Proof state:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
φ ψ : MvPowerSeries σ R
a : R
this : ∀ p ∈ antidiagonal m, (coeff R p.1) φ * (coeff R p.2) ((monomial R n) a) ≠ 0 → p.2 = n
⊢ (if n ≤ m then ∑ x in {(m - n, n)}, (coeff R x.1) φ * (coeff R x.2) ((monomial R n) a)
    else ∑ x in ∅, (coeff R x.1) φ * (coeff R x.2) ((monomial R n) a)) =
    if n ≤ m then (coeff R (m - n)) φ * a else 0

Next tactic:
simp only [Finset.sum_singleton, coeff_monomial_same, Finset.sum_empty]
theorem coeff_add_monomial_mul (a : R) :
    coeff R (m + n) (monomial R m a * φ) = a * coeff R n φ := by
  

Proof state:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
φ ψ : MvPowerSeries σ R
a : R
⊢ (coeff R (m + n)) ((monomial R m) a * φ) = a * (coeff R n) φ

Next tactic:
rw [coeff_monomial_mul, if_pos, add_tsub_cancel_left]
theorem coeff_add_monomial_mul (a : R) :
    coeff R (m + n) (monomial R m a * φ) = a * coeff R n φ := by
  rw [coeff_monomial_mul, if_pos, add_tsub_cancel_left]
  

Proof state:
case hc
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
φ ψ : MvPowerSeries σ R
a : R
⊢ m ≤ m + n

Next tactic:
exact le_add_right le_rfl
theorem coeff_add_mul_monomial (a : R) :
    coeff R (m + n) (φ * monomial R n a) = coeff R m φ * a := by
  

Proof state:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
φ ψ : MvPowerSeries σ R
a : R
⊢ (coeff R (m + n)) (φ * (monomial R n) a) = (coeff R m) φ * a

Next tactic:
rw [coeff_mul_monomial, if_pos, add_tsub_cancel_right]
theorem coeff_add_mul_monomial (a : R) :
    coeff R (m + n) (φ * monomial R n a) = coeff R m φ * a := by
  rw [coeff_mul_monomial, if_pos, add_tsub_cancel_right]
  

Proof state:
case hc
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
φ ψ : MvPowerSeries σ R
a : R
⊢ n ≤ m + n

Next tactic:
exact le_add_left le_rfl
@[simp]
theorem commute_monomial {a : R} {n} :
    Commute φ (monomial R n a) ↔ ∀ m, Commute (coeff R m φ) a := by
  

Proof state:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n✝ : σ →₀ ℕ
φ ψ : MvPowerSeries σ R
a : R
n : σ →₀ ℕ
⊢ Commute φ ((monomial R n) a) ↔ ∀ (m : σ →₀ ℕ), Commute ((coeff R m) φ) a

Next tactic:
refine' ext_iff.trans ⟨fun h m => _, fun h m => _⟩
@[simp]
theorem commute_monomial {a : R} {n} :
    Commute φ (monomial R n a) ↔ ∀ m, Commute (coeff R m φ) a := by
  refine' ext_iff.trans ⟨fun h m => _, fun h m => _⟩
  · 

Proof state:
case refine'_1
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m✝ n✝ : σ →₀ ℕ
φ ψ : MvPowerSeries σ R
a : R
n : σ →₀ ℕ
h : ∀ (n_1 : σ →₀ ℕ), (coeff R n_1) (φ * (monomial R n) a) = (coeff R n_1) ((monomial R n) a * φ)
m : σ →₀ ℕ
⊢ Commute ((coeff R m) φ) a

Next tactic:
have := h (m + n)
@[simp]
theorem commute_monomial {a : R} {n} :
    Commute φ (monomial R n a) ↔ ∀ m, Commute (coeff R m φ) a := by
  refine' ext_iff.trans ⟨fun h m => _, fun h m => _⟩
  · have := h (m + n)
    

Proof state:
case refine'_1
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m✝ n✝ : σ →₀ ℕ
φ ψ : MvPowerSeries σ R
a : R
n : σ →₀ ℕ
h : ∀ (n_1 : σ →₀ ℕ), (coeff R n_1) (φ * (monomial R n) a) = (coeff R n_1) ((monomial R n) a * φ)
m : σ →₀ ℕ
this : (coeff R (m + n)) (φ * (monomial R n) a) = (coeff R (m + n)) ((monomial R n) a * φ)
⊢ Commute ((coeff R m) φ) a

Next tactic:
rwa [coeff_add_mul_monomial, add_comm, coeff_add_monomial_mul] at this
@[simp]
theorem commute_monomial {a : R} {n} :
    Commute φ (monomial R n a) ↔ ∀ m, Commute (coeff R m φ) a := by
  refine' ext_iff.trans ⟨fun h m => _, fun h m => _⟩
  · have := h (m + n)
    rwa [coeff_add_mul_monomial, add_comm, coeff_add_monomial_mul] at this
  · 

Proof state:
case refine'_2
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m✝ n✝ : σ →₀ ℕ
φ ψ : MvPowerSeries σ R
a : R
n : σ →₀ ℕ
h : ∀ (m : σ →₀ ℕ), Commute ((coeff R m) φ) a
m : σ →₀ ℕ
⊢ (coeff R m) (φ * (monomial R n) a) = (coeff R m) ((monomial R n) a * φ)

Next tactic:
rw [coeff_mul_monomial, coeff_monomial_mul]
@[simp]
theorem commute_monomial {a : R} {n} :
    Commute φ (monomial R n a) ↔ ∀ m, Commute (coeff R m φ) a := by
  refine' ext_iff.trans ⟨fun h m => _, fun h m => _⟩
  · have := h (m + n)
    rwa [coeff_add_mul_monomial, add_comm, coeff_add_monomial_mul] at this
  · rw [coeff_mul_monomial, coeff_monomial_mul]
    

Proof state:
case refine'_2
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m✝ n✝ : σ →₀ ℕ
φ ψ : MvPowerSeries σ R
a : R
n : σ →₀ ℕ
h : ∀ (m : σ →₀ ℕ), Commute ((coeff R m) φ) a
m : σ →₀ ℕ
⊢ (if n ≤ m then (coeff R (m - n)) φ * a else 0) = if n ≤ m then a * (coeff R (m - n)) φ else 0

Next tactic:
split_ifs <;> [apply h; rfl]
@[simp]
theorem commute_monomial {a : R} {n} :
    Commute φ (monomial R n a) ↔ ∀ m, Commute (coeff R m φ) a := by
  refine' ext_iff.trans ⟨fun h m => _, fun h m => _⟩
  · have := h (m + n)
    rwa [coeff_add_mul_monomial, add_comm, coeff_add_monomial_mul] at this
  · rw [coeff_mul_monomial, coeff_monomial_mul]
    

Proof state:
case refine'_2
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m✝ n✝ : σ →₀ ℕ
φ ψ : MvPowerSeries σ R
a : R
n : σ →₀ ℕ
h : ∀ (m : σ →₀ ℕ), Commute ((coeff R m) φ) a
m : σ →₀ ℕ
⊢ (if n ≤ m then (coeff R (m - n)) φ * a else 0) = if n ≤ m then a * (coeff R (m - n)) φ else 0

Next tactic:
split_ifs
@[simp]
theorem commute_monomial {a : R} {n} :
    Commute φ (monomial R n a) ↔ ∀ m, Commute (coeff R m φ) a := by
  refine' ext_iff.trans ⟨fun h m => _, fun h m => _⟩
  · have := h (m + n)
    rwa [coeff_add_mul_monomial, add_comm, coeff_add_monomial_mul] at this
  · rw [coeff_mul_monomial, coeff_monomial_mul]
    split_ifs <;> [

Proof state:
case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m✝ n✝ : σ →₀ ℕ
φ ψ : MvPowerSeries σ R
a : R
n : σ →₀ ℕ
h : ∀ (m : σ →₀ ℕ), Commute ((coeff R m) φ) a
m : σ →₀ ℕ
h✝ : n ≤ m
⊢ (coeff R (m - n)) φ * a = a * (coeff R (m - n)) φ

Next tactic:
apply h
@[simp]
theorem commute_monomial {a : R} {n} :
    Commute φ (monomial R n a) ↔ ∀ m, Commute (coeff R m φ) a := by
  refine' ext_iff.trans ⟨fun h m => _, fun h m => _⟩
  · have := h (m + n)
    rwa [coeff_add_mul_monomial, add_comm, coeff_add_monomial_mul] at this
  · rw [coeff_mul_monomial, coeff_monomial_mul]
    split_ifs <;> [apply h; 

Proof state:
case neg
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m✝ n✝ : σ →₀ ℕ
φ ψ : MvPowerSeries σ R
a : R
n : σ →₀ ℕ
h : ∀ (m : σ →₀ ℕ), Commute ((coeff R m) φ) a
m : σ →₀ ℕ
h✝ : ¬n ≤ m
⊢ 0 = 0

Next tactic:
rfl
protected theorem one_mul : (1 : MvPowerSeries σ R) * φ = φ :=
  ext fun n => by 

Proof state:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n✝ : σ →₀ ℕ
φ ψ : MvPowerSeries σ R
n : σ →₀ ℕ
⊢ (coeff R n) (1 * φ) = (coeff R n) φ

Next tactic:
simpa using coeff_add_monomial_mul 0 n φ 1
protected theorem mul_one : φ * 1 = φ :=
  ext fun n => by 

Proof state:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n✝ : σ →₀ ℕ
φ ψ : MvPowerSeries σ R
n : σ →₀ ℕ
⊢ (coeff R n) (φ * 1) = (coeff R n) φ

Next tactic:
simpa using coeff_add_mul_monomial n 0 φ 1
protected theorem mul_add (φ₁ φ₂ φ₃ : MvPowerSeries σ R) : φ₁ * (φ₂ + φ₃) = φ₁ * φ₂ + φ₁ * φ₃ :=
  ext fun n => by
    

Proof state:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n✝ : σ →₀ ℕ
φ ψ φ₁ φ₂ φ₃ : MvPowerSeries σ R
n : σ →₀ ℕ
⊢ (coeff R n) (φ₁ * (φ₂ + φ₃)) = (coeff R n) (φ₁ * φ₂ + φ₁ * φ₃)

Next tactic:
classical simp only [coeff_mul, mul_add, Finset.sum_add_distrib, LinearMap.map_add]
protected theorem mul_add (φ₁ φ₂ φ₃ : MvPowerSeries σ R) : φ₁ * (φ₂ + φ₃) = φ₁ * φ₂ + φ₁ * φ₃ :=
  ext fun n => by
    classical 

Proof state:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n✝ : σ →₀ ℕ
φ ψ φ₁ φ₂ φ₃ : MvPowerSeries σ R
n : σ →₀ ℕ
⊢ (coeff R n) (φ₁ * (φ₂ + φ₃)) = (coeff R n) (φ₁ * φ₂ + φ₁ * φ₃)

Next tactic:
simp only [coeff_mul, mul_add, Finset.sum_add_distrib, LinearMap.map_add]
protected theorem add_mul (φ₁ φ₂ φ₃ : MvPowerSeries σ R) : (φ₁ + φ₂) * φ₃ = φ₁ * φ₃ + φ₂ * φ₃ :=
  ext fun n => by
    

Proof state:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n✝ : σ →₀ ℕ
φ ψ φ₁ φ₂ φ₃ : MvPowerSeries σ R
n : σ →₀ ℕ
⊢ (coeff R n) ((φ₁ + φ₂) * φ₃) = (coeff R n) (φ₁ * φ₃ + φ₂ * φ₃)

Next tactic:
classical simp only [coeff_mul, add_mul, Finset.sum_add_distrib, LinearMap.map_add]
protected theorem add_mul (φ₁ φ₂ φ₃ : MvPowerSeries σ R) : (φ₁ + φ₂) * φ₃ = φ₁ * φ₃ + φ₂ * φ₃ :=
  ext fun n => by
    classical 

Proof state:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n✝ : σ →₀ ℕ
φ ψ φ₁ φ₂ φ₃ : MvPowerSeries σ R
n : σ →₀ ℕ
⊢ (coeff R n) ((φ₁ + φ₂) * φ₃) = (coeff R n) (φ₁ * φ₃ + φ₂ * φ₃)

Next tactic:
simp only [coeff_mul, add_mul, Finset.sum_add_distrib, LinearMap.map_add]
protected theorem mul_assoc (φ₁ φ₂ φ₃ : MvPowerSeries σ R) : φ₁ * φ₂ * φ₃ = φ₁ * (φ₂ * φ₃) := by
  

Proof state:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
φ ψ φ₁ φ₂ φ₃ : MvPowerSeries σ R
⊢ φ₁ * φ₂ * φ₃ = φ₁ * (φ₂ * φ₃)

Next tactic:
ext1 n
protected theorem mul_assoc (φ₁ φ₂ φ₃ : MvPowerSeries σ R) : φ₁ * φ₂ * φ₃ = φ₁ * (φ₂ * φ₃) := by
  ext1 n
  

Proof state:
case h
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n✝ : σ →₀ ℕ
φ ψ φ₁ φ₂ φ₃ : MvPowerSeries σ R
n : σ →₀ ℕ
⊢ (coeff R n) (φ₁ * φ₂ * φ₃) = (coeff R n) (φ₁ * (φ₂ * φ₃))

Next tactic:
classical
  simp only [coeff_mul, Finset.sum_mul, Finset.mul_sum, Finset.sum_sigma']
  refine' Finset.sum_bij (fun p _ => ⟨(p.2.1, p.2.2 + p.1.2), (p.2.2, p.1.2)⟩) _ _ _ _ <;>
    simp only [mem_antidiagonal, Finset.mem_sigma, heq_iff_eq, Prod.mk.inj_iff, and_imp,
      exists_prop]
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩
    dsimp only
    rintro rfl rfl
    simp [add_assoc]
  · rintro ⟨⟨a, b⟩, ⟨c, d⟩⟩
    dsimp only
    rintro rfl rfl
    apply mul_assoc
  · rintro ⟨⟨a, b⟩, ⟨c, d⟩⟩ ⟨⟨i, j⟩, ⟨k, l⟩⟩
    dsimp only
    rintro rfl rfl - rfl
    simp only [Sigma.mk.inj_iff, Prod.mk.injEq, heq_iff_eq, and_imp]
    rintro rfl - rfl rfl
    simp only [and_self]
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩
    dsimp only
    rintro rfl rfl
    refine' ⟨⟨(i + k, l), (i, k)⟩, _, _⟩ <;> simp [add_assoc]
protected theorem mul_assoc (φ₁ φ₂ φ₃ : MvPowerSeries σ R) : φ₁ * φ₂ * φ₃ = φ₁ * (φ₂ * φ₃) := by
  ext1 n
  classical
  

Proof state:
case h
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n✝ : σ →₀ ℕ
φ ψ φ₁ φ₂ φ₃ : MvPowerSeries σ R
n : σ →₀ ℕ
⊢ (coeff R n) (φ₁ * φ₂ * φ₃) = (coeff R n) (φ₁ * (φ₂ * φ₃))

Next tactic:
simp only [coeff_mul, Finset.sum_mul, Finset.mul_sum, Finset.sum_sigma']
protected theorem mul_assoc (φ₁ φ₂ φ₃ : MvPowerSeries σ R) : φ₁ * φ₂ * φ₃ = φ₁ * (φ₂ * φ₃) := by
  ext1 n
  classical
  simp only [coeff_mul, Finset.sum_mul, Finset.mul_sum, Finset.sum_sigma']
  

Proof state:
case h
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n✝ : σ →₀ ℕ
φ ψ φ₁ φ₂ φ₃ : MvPowerSeries σ R
n : σ →₀ ℕ
⊢ ∑ x in Finset.sigma (antidiagonal n) fun a => antidiagonal a.1,
      (coeff R x.snd.1) φ₁ * (coeff R x.snd.2) φ₂ * (coeff R x.fst.2) φ₃ =
    ∑ x in Finset.sigma (antidiagonal n) fun a => antidiagonal a.2,
      (coeff R x.fst.1) φ₁ * ((coeff R x.snd.1) φ₂ * (coeff R x.snd.2) φ₃)

Next tactic:
refine' Finset.sum_bij (fun p _ => ⟨(p.2.1, p.2.2 + p.1.2), (p.2.2, p.1.2)⟩) _ _ _ _
protected theorem mul_assoc (φ₁ φ₂ φ₃ : MvPowerSeries σ R) : φ₁ * φ₂ * φ₃ = φ₁ * (φ₂ * φ₃) := by
  ext1 n
  classical
  simp only [coeff_mul, Finset.sum_mul, Finset.mul_sum, Finset.sum_sigma']
  refine' Finset.sum_bij (fun p _ => ⟨(p.2.1, p.2.2 + p.1.2), (p.2.2, p.1.2)⟩) _ _ _ _ <;>
    

Proof state:
case h.refine'_1
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n✝ : σ →₀ ℕ
φ ψ φ₁ φ₂ φ₃ : MvPowerSeries σ R
n : σ →₀ ℕ
⊢ ∀ (a : (_ : (σ →₀ ℕ) × (σ →₀ ℕ)) × (σ →₀ ℕ) × (σ →₀ ℕ))
    (ha : a ∈ Finset.sigma (antidiagonal n) fun a => antidiagonal a.1),
    (fun p x => { fst := (p.snd.1, p.snd.2 + p.fst.2), snd := (p.snd.2, p.fst.2) }) a ha ∈
      Finset.sigma (antidiagonal n) fun a => antidiagonal a.2

Next tactic:
simp only [mem_antidiagonal, Finset.mem_sigma, heq_iff_eq, Prod.mk.inj_iff, and_imp,
      exists_prop]
protected theorem mul_assoc (φ₁ φ₂ φ₃ : MvPowerSeries σ R) : φ₁ * φ₂ * φ₃ = φ₁ * (φ₂ * φ₃) := by
  ext1 n
  classical
  simp only [coeff_mul, Finset.sum_mul, Finset.mul_sum, Finset.sum_sigma']
  refine' Finset.sum_bij (fun p _ => ⟨(p.2.1, p.2.2 + p.1.2), (p.2.2, p.1.2)⟩) _ _ _ _ <;>
    

Proof state:
case h.refine'_2
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n✝ : σ →₀ ℕ
φ ψ φ₁ φ₂ φ₃ : MvPowerSeries σ R
n : σ →₀ ℕ
⊢ ∀ (a : (_ : (σ →₀ ℕ) × (σ →₀ ℕ)) × (σ →₀ ℕ) × (σ →₀ ℕ))
    (ha : a ∈ Finset.sigma (antidiagonal n) fun a => antidiagonal a.1),
    (coeff R a.snd.1) φ₁ * (coeff R a.snd.2) φ₂ * (coeff R a.fst.2) φ₃ =
      (coeff R ((fun p x => { fst := (p.snd.1, p.snd.2 + p.fst.2), snd := (p.snd.2, p.fst.2) }) a ha).fst.1) φ₁ *
        ((coeff R ((fun p x => { fst := (p.snd.1, p.snd.2 + p.fst.2), snd := (p.snd.2, p.fst.2) }) a ha).snd.1) φ₂ *
          (coeff R ((fun p x => { fst := (p.snd.1, p.snd.2 + p.fst.2), snd := (p.snd.2, p.fst.2) }) a ha).snd.2) φ₃)

Next tactic:
simp only [mem_antidiagonal, Finset.mem_sigma, heq_iff_eq, Prod.mk.inj_iff, and_imp,
      exists_prop]
protected theorem mul_assoc (φ₁ φ₂ φ₃ : MvPowerSeries σ R) : φ₁ * φ₂ * φ₃ = φ₁ * (φ₂ * φ₃) := by
  ext1 n
  classical
  simp only [coeff_mul, Finset.sum_mul, Finset.mul_sum, Finset.sum_sigma']
  refine' Finset.sum_bij (fun p _ => ⟨(p.2.1, p.2.2 + p.1.2), (p.2.2, p.1.2)⟩) _ _ _ _ <;>
    

Proof state:
case h.refine'_3
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n✝ : σ →₀ ℕ
φ ψ φ₁ φ₂ φ₃ : MvPowerSeries σ R
n : σ →₀ ℕ
⊢ ∀ (a₁ a₂ : (_ : (σ →₀ ℕ) × (σ →₀ ℕ)) × (σ →₀ ℕ) × (σ →₀ ℕ))
    (ha₁ : a₁ ∈ Finset.sigma (antidiagonal n) fun a => antidiagonal a.1)
    (ha₂ : a₂ ∈ Finset.sigma (antidiagonal n) fun a => antidiagonal a.1),
    (fun p x => { fst := (p.snd.1, p.snd.2 + p.fst.2), snd := (p.snd.2, p.fst.2) }) a₁ ha₁ =
        (fun p x => { fst := (p.snd.1, p.snd.2 + p.fst.2), snd := (p.snd.2, p.fst.2) }) a₂ ha₂ →
      a₁ = a₂

Next tactic:
simp only [mem_antidiagonal, Finset.mem_sigma, heq_iff_eq, Prod.mk.inj_iff, and_imp,
      exists_prop]
protected theorem mul_assoc (φ₁ φ₂ φ₃ : MvPowerSeries σ R) : φ₁ * φ₂ * φ₃ = φ₁ * (φ₂ * φ₃) := by
  ext1 n
  classical
  simp only [coeff_mul, Finset.sum_mul, Finset.mul_sum, Finset.sum_sigma']
  refine' Finset.sum_bij (fun p _ => ⟨(p.2.1, p.2.2 + p.1.2), (p.2.2, p.1.2)⟩) _ _ _ _ <;>
    

Proof state:
case h.refine'_4
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n✝ : σ →₀ ℕ
φ ψ φ₁ φ₂ φ₃ : MvPowerSeries σ R
n : σ →₀ ℕ
⊢ ∀ b ∈ Finset.sigma (antidiagonal n) fun a => antidiagonal a.2,
    ∃ a,
      ∃ (ha : a ∈ Finset.sigma (antidiagonal n) fun a => antidiagonal a.1),
        b = (fun p x => { fst := (p.snd.1, p.snd.2 + p.fst.2), snd := (p.snd.2, p.fst.2) }) a ha

Next tactic:
simp only [mem_antidiagonal, Finset.mem_sigma, heq_iff_eq, Prod.mk.inj_iff, and_imp,
      exists_prop]
protected theorem mul_assoc (φ₁ φ₂ φ₃ : MvPowerSeries σ R) : φ₁ * φ₂ * φ₃ = φ₁ * (φ₂ * φ₃) := by
  ext1 n
  classical
  simp only [coeff_mul, Finset.sum_mul, Finset.mul_sum, Finset.sum_sigma']
  refine' Finset.sum_bij (fun p _ => ⟨(p.2.1, p.2.2 + p.1.2), (p.2.2, p.1.2)⟩) _ _ _ _ <;>
    simp only [mem_antidiagonal, Finset.mem_sigma, heq_iff_eq, Prod.mk.inj_iff, and_imp,
      exists_prop]
  · 

Proof state:
case h.refine'_1
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n✝ : σ →₀ ℕ
φ ψ φ₁ φ₂ φ₃ : MvPowerSeries σ R
n : σ →₀ ℕ
⊢ ∀ (a : (_ : (σ →₀ ℕ) × (σ →₀ ℕ)) × (σ →₀ ℕ) × (σ →₀ ℕ)),
    a.fst.1 + a.fst.2 = n → a.snd.1 + a.snd.2 = a.fst.1 → a.snd.1 + (a.snd.2 + a.fst.2) = n ∧ True

Next tactic:
rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩
protected theorem mul_assoc (φ₁ φ₂ φ₃ : MvPowerSeries σ R) : φ₁ * φ₂ * φ₃ = φ₁ * (φ₂ * φ₃) := by
  ext1 n
  classical
  simp only [coeff_mul, Finset.sum_mul, Finset.mul_sum, Finset.sum_sigma']
  refine' Finset.sum_bij (fun p _ => ⟨(p.2.1, p.2.2 + p.1.2), (p.2.2, p.1.2)⟩) _ _ _ _ <;>
    simp only [mem_antidiagonal, Finset.mem_sigma, heq_iff_eq, Prod.mk.inj_iff, and_imp,
      exists_prop]
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩
    

Proof state:
case h.refine'_1.mk.mk.mk
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n✝ : σ →₀ ℕ
φ ψ φ₁ φ₂ φ₃ : MvPowerSeries σ R
n i j k l : σ →₀ ℕ
⊢ { fst := (i, j), snd := (k, l) }.fst.1 + { fst := (i, j), snd := (k, l) }.fst.2 = n →
    { fst := (i, j), snd := (k, l) }.snd.1 + { fst := (i, j), snd := (k, l) }.snd.2 =
        { fst := (i, j), snd := (k, l) }.fst.1 →
      { fst := (i, j), snd := (k, l) }.snd.1 +
            ({ fst := (i, j), snd := (k, l) }.snd.2 + { fst := (i, j), snd := (k, l) }.fst.2) =
          n ∧
        True

Next tactic:
dsimp only
protected theorem mul_assoc (φ₁ φ₂ φ₃ : MvPowerSeries σ R) : φ₁ * φ₂ * φ₃ = φ₁ * (φ₂ * φ₃) := by
  ext1 n
  classical
  simp only [coeff_mul, Finset.sum_mul, Finset.mul_sum, Finset.sum_sigma']
  refine' Finset.sum_bij (fun p _ => ⟨(p.2.1, p.2.2 + p.1.2), (p.2.2, p.1.2)⟩) _ _ _ _ <;>
    simp only [mem_antidiagonal, Finset.mem_sigma, heq_iff_eq, Prod.mk.inj_iff, and_imp,
      exists_prop]
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩
    dsimp only
    

Proof state:
case h.refine'_1.mk.mk.mk
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n✝ : σ →₀ ℕ
φ ψ φ₁ φ₂ φ₃ : MvPowerSeries σ R
n i j k l : σ →₀ ℕ
⊢ i + j = n → k + l = i → k + (l + j) = n ∧ True

Next tactic:
rintro rfl rfl
protected theorem mul_assoc (φ₁ φ₂ φ₃ : MvPowerSeries σ R) : φ₁ * φ₂ * φ₃ = φ₁ * (φ₂ * φ₃) := by
  ext1 n
  classical
  simp only [coeff_mul, Finset.sum_mul, Finset.mul_sum, Finset.sum_sigma']
  refine' Finset.sum_bij (fun p _ => ⟨(p.2.1, p.2.2 + p.1.2), (p.2.2, p.1.2)⟩) _ _ _ _ <;>
    simp only [mem_antidiagonal, Finset.mem_sigma, heq_iff_eq, Prod.mk.inj_iff, and_imp,
      exists_prop]
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩
    dsimp only
    rintro rfl rfl
    

Proof state:
case h.refine'_1.mk.mk.mk
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
φ ψ φ₁ φ₂ φ₃ : MvPowerSeries σ R
j k l : σ →₀ ℕ
⊢ k + (l + j) = k + l + j ∧ True

Next tactic:
simp [add_assoc]
protected theorem mul_assoc (φ₁ φ₂ φ₃ : MvPowerSeries σ R) : φ₁ * φ₂ * φ₃ = φ₁ * (φ₂ * φ₃) := by
  ext1 n
  classical
  simp only [coeff_mul, Finset.sum_mul, Finset.mul_sum, Finset.sum_sigma']
  refine' Finset.sum_bij (fun p _ => ⟨(p.2.1, p.2.2 + p.1.2), (p.2.2, p.1.2)⟩) _ _ _ _ <;>
    simp only [mem_antidiagonal, Finset.mem_sigma, heq_iff_eq, Prod.mk.inj_iff, and_imp,
      exists_prop]
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩
    dsimp only
    rintro rfl rfl
    simp [add_assoc]
  · 

Proof state:
case h.refine'_2
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n✝ : σ →₀ ℕ
φ ψ φ₁ φ₂ φ₃ : MvPowerSeries σ R
n : σ →₀ ℕ
⊢ ∀ (a : (_ : (σ →₀ ℕ) × (σ →₀ ℕ)) × (σ →₀ ℕ) × (σ →₀ ℕ)),
    a.fst.1 + a.fst.2 = n →
      a.snd.1 + a.snd.2 = a.fst.1 →
        (coeff R a.snd.1) φ₁ * (coeff R a.snd.2) φ₂ * (coeff R a.fst.2) φ₃ =
          (coeff R a.snd.1) φ₁ * ((coeff R a.snd.2) φ₂ * (coeff R a.fst.2) φ₃)

Next tactic:
rintro ⟨⟨a, b⟩, ⟨c, d⟩⟩
protected theorem mul_assoc (φ₁ φ₂ φ₃ : MvPowerSeries σ R) : φ₁ * φ₂ * φ₃ = φ₁ * (φ₂ * φ₃) := by
  ext1 n
  classical
  simp only [coeff_mul, Finset.sum_mul, Finset.mul_sum, Finset.sum_sigma']
  refine' Finset.sum_bij (fun p _ => ⟨(p.2.1, p.2.2 + p.1.2), (p.2.2, p.1.2)⟩) _ _ _ _ <;>
    simp only [mem_antidiagonal, Finset.mem_sigma, heq_iff_eq, Prod.mk.inj_iff, and_imp,
      exists_prop]
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩
    dsimp only
    rintro rfl rfl
    simp [add_assoc]
  · rintro ⟨⟨a, b⟩, ⟨c, d⟩⟩
    

Proof state:
case h.refine'_2.mk.mk.mk
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n✝ : σ →₀ ℕ
φ ψ φ₁ φ₂ φ₃ : MvPowerSeries σ R
n a b c d : σ →₀ ℕ
⊢ { fst := (a, b), snd := (c, d) }.fst.1 + { fst := (a, b), snd := (c, d) }.fst.2 = n →
    { fst := (a, b), snd := (c, d) }.snd.1 + { fst := (a, b), snd := (c, d) }.snd.2 =
        { fst := (a, b), snd := (c, d) }.fst.1 →
      (coeff R { fst := (a, b), snd := (c, d) }.snd.1) φ₁ * (coeff R { fst := (a, b), snd := (c, d) }.snd.2) φ₂ *
          (coeff R { fst := (a, b), snd := (c, d) }.fst.2) φ₃ =
        (coeff R { fst := (a, b), snd := (c, d) }.snd.1) φ₁ *
          ((coeff R { fst := (a, b), snd := (c, d) }.snd.2) φ₂ * (coeff R { fst := (a, b), snd := (c, d) }.fst.2) φ₃)

Next tactic:
dsimp only
protected theorem mul_assoc (φ₁ φ₂ φ₃ : MvPowerSeries σ R) : φ₁ * φ₂ * φ₃ = φ₁ * (φ₂ * φ₃) := by
  ext1 n
  classical
  simp only [coeff_mul, Finset.sum_mul, Finset.mul_sum, Finset.sum_sigma']
  refine' Finset.sum_bij (fun p _ => ⟨(p.2.1, p.2.2 + p.1.2), (p.2.2, p.1.2)⟩) _ _ _ _ <;>
    simp only [mem_antidiagonal, Finset.mem_sigma, heq_iff_eq, Prod.mk.inj_iff, and_imp,
      exists_prop]
  · rintro ⟨⟨i, j⟩, ⟨k, l⟩⟩
    dsimp only
    rintro rfl rfl
    simp [add_assoc]
  · rintro ⟨⟨a, b⟩, ⟨c, d⟩⟩
    dsimp only
    

Proof state:
case h.refine'_2.mk.mk.mk
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n✝ : σ →₀ ℕ
φ ψ φ₁ φ₂ φ₃ : MvPowerSeries σ R
n a b c d : σ →₀ ℕ
⊢ a + b = n →
    c + d = a → (coeff R c) φ₁ * (coeff R d) φ₂ * (coeff R b) φ₃ = (coeff R c) φ₁ * ((coeff R d) φ₂ * (coeff R b) φ₃)

Next tactic:
rintro rfl rfl