l3lab/ntp-mathlib huggingface dataset:
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
    rintro rfl rfl
    

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
b c d : σ →₀ ℕ
⊢ (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:
apply mul_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]
  · rintro ⟨⟨a, b⟩, ⟨c, d⟩⟩
    dsimp only
    rintro rfl rfl
    apply mul_assoc
  · 

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)),
    a₁.fst.1 + a₁.fst.2 = n →
      a₁.snd.1 + a₁.snd.2 = a₁.fst.1 →
        a₂.fst.1 + a₂.fst.2 = n →
          a₂.snd.1 + a₂.snd.2 = a₂.fst.1 →
            { fst := (a₁.snd.1, a₁.snd.2 + a₁.fst.2), snd := (a₁.snd.2, a₁.fst.2) } =
                { fst := (a₂.snd.1, a₂.snd.2 + a₂.fst.2), snd := (a₂.snd.2, a₂.fst.2) } →
              a₁ = a₂

Next tactic:
rintro ⟨⟨a, b⟩, ⟨c, d⟩⟩ ⟨⟨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⟩⟩
    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⟩⟩
    

Proof state:
case h.refine'_3.mk.mk.mk.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 i j k l : 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 →
      { 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 :=
                  ({ fst := (a, b), snd := (c, d) }.snd.1,
                    { fst := (a, b), snd := (c, d) }.snd.2 + { fst := (a, b), snd := (c, d) }.fst.2),
                snd := ({ fst := (a, b), snd := (c, d) }.snd.2, { fst := (a, b), snd := (c, d) }.fst.2) } =
              {
                fst :=
                  ({ fst := (i, j), snd := (k, l) }.snd.1,
                    { fst := (i, j), snd := (k, l) }.snd.2 + { fst := (i, j), snd := (k, l) }.fst.2),
                snd := ({ fst := (i, j), snd := (k, l) }.snd.2, { fst := (i, j), snd := (k, l) }.fst.2) } →
            { fst := (a, b), snd := (c, d) } = { fst := (i, j), snd := (k, l) }

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
    rintro rfl rfl
    apply mul_assoc
  · rintro ⟨⟨a, b⟩, ⟨c, d⟩⟩ ⟨⟨i, j⟩, ⟨k, l⟩⟩
    dsimp only
    

Proof state:
case h.refine'_3.mk.mk.mk.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 i j k l : σ →₀ ℕ
⊢ a + b = n →
    c + d = a →
      i + j = n →
        k + l = i →
          { 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:
rintro rfl 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
    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
    

Proof state:
case h.refine'_3.mk.mk.mk.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
b c d j k l : σ →₀ ℕ
⊢ { fst := (c, d + b), snd := (d, b) } = { fst := (k, l + j), snd := (l, j) } →
    { fst := (c + d, b), snd := (c, d) } = { fst := (k + l, j), snd := (k, l) }

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

Proof state:
case h.refine'_3.mk.mk.mk.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
b c d j k l : σ →₀ ℕ
⊢ c = k → d + b = l + j → d = l → b = j → (c + d = k + l ∧ b = j) ∧ c = k ∧ d = l

Next tactic:
rintro rfl - 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
    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
    

Proof state:
case h.refine'_3.mk.mk.mk.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
b c d : σ →₀ ℕ
⊢ (c + d = c + d ∧ b = b) ∧ c = c ∧ d = d

Next tactic:
simp only [and_self]
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
    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]
  · 

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 : (_ : (σ →₀ ℕ) × (σ →₀ ℕ)) × (σ →₀ ℕ) × (σ →₀ ℕ)),
    b.fst.1 + b.fst.2 = n →
      b.snd.1 + b.snd.2 = b.fst.2 →
        ∃ a,
          (a.fst.1 + a.fst.2 = n ∧ a.snd.1 + a.snd.2 = a.fst.1) ∧
            b = { fst := (a.snd.1, a.snd.2 + a.fst.2), snd := (a.snd.2, a.fst.2) }

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⟩⟩
    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⟩⟩
    

Proof state:
case h.refine'_4.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.2 →
      ∃ a,
        (a.fst.1 + a.fst.2 = n ∧ a.snd.1 + a.snd.2 = a.fst.1) ∧
          { fst := (i, j), snd := (k, l) } = { fst := (a.snd.1, a.snd.2 + a.fst.2), snd := (a.snd.2, a.fst.2) }

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
    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
    

Proof state:
case h.refine'_4.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 = j →
      ∃ a,
        (a.fst.1 + a.fst.2 = n ∧ a.snd.1 + a.snd.2 = a.fst.1) ∧
          { fst := (i, j), snd := (k, l) } = { fst := (a.snd.1, a.snd.2 + a.fst.2), snd := (a.snd.2, a.fst.2) }

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
    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
    

Proof state:
case h.refine'_4.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
i k l : σ →₀ ℕ
⊢ ∃ a,
    (a.fst.1 + a.fst.2 = i + (k + l) ∧ a.snd.1 + a.snd.2 = a.fst.1) ∧
      { fst := (i, k + l), snd := (k, l) } = { fst := (a.snd.1, a.snd.2 + a.fst.2), snd := (a.snd.2, a.fst.2) }

Next tactic:
refine' ⟨⟨(i + k, l), (i, k)⟩, _, _⟩
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
    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)⟩, _, _⟩ <;> 

Proof state:
case h.refine'_4.mk.mk.mk.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
i k l : σ →₀ ℕ
⊢ { fst := (i + k, l), snd := (i, k) }.fst.1 + { fst := (i + k, l), snd := (i, k) }.fst.2 = i + (k + l) ∧
    { fst := (i + k, l), snd := (i, k) }.snd.1 + { fst := (i + k, l), snd := (i, k) }.snd.2 =
      { fst := (i + k, l), snd := (i, k) }.fst.1

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]
  · 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)⟩, _, _⟩ <;> 

Proof state:
case h.refine'_4.mk.mk.mk.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
i k l : σ →₀ ℕ
⊢ { fst := (i, k + l), snd := (k, l) } =
    {
      fst :=
        ({ fst := (i + k, l), snd := (i, k) }.snd.1,
          { fst := (i + k, l), snd := (i, k) }.snd.2 + { fst := (i + k, l), snd := (i, k) }.fst.2),
      snd := ({ fst := (i + k, l), snd := (i, k) }.snd.2, { fst := (i + k, l), snd := (i, k) }.fst.2) }

Next tactic:
simp [add_assoc]
instance [CommSemiring R] : CommSemiring (MvPowerSeries σ R) :=
  { show Semiring (MvPowerSeries σ R) by 

Proof state:
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
⊢ Semiring (MvPowerSeries σ R)

Next tactic:
infer_instance
instance [CommSemiring R] : CommSemiring (MvPowerSeries σ R) :=
  { show Semiring (MvPowerSeries σ R) by infer_instance with
    mul_comm := fun φ ψ =>
      ext fun n => by
        

Proof state:
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
src✝ : Semiring (MvPowerSeries σ R) :=
  let_fun this := inferInstance;
  this
φ ψ : MvPowerSeries σ R
n : σ →₀ ℕ
⊢ (coeff R n) (φ * ψ) = (coeff R n) (ψ * φ)

Next tactic:
classical
        simpa only [coeff_mul, mul_comm] using
          sum_antidiagonal_swap n fun a b => coeff R a φ * coeff R b ψ
instance [CommSemiring R] : CommSemiring (MvPowerSeries σ R) :=
  { show Semiring (MvPowerSeries σ R) by infer_instance with
    mul_comm := fun φ ψ =>
      ext fun n => by
        classical
        

Proof state:
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
src✝ : Semiring (MvPowerSeries σ R) :=
  let_fun this := inferInstance;
  this
φ ψ : MvPowerSeries σ R
n : σ →₀ ℕ
⊢ (coeff R n) (φ * ψ) = (coeff R n) (ψ * φ)

Next tactic:
simpa only [coeff_mul, mul_comm] using
          sum_antidiagonal_swap n fun a b => coeff R a φ * coeff R b ψ
theorem monomial_mul_monomial (m n : σ →₀ ℕ) (a b : R) :
    monomial R m a * monomial R n b = monomial R (m + n) (a * b) := by
  

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

Next tactic:
classical
  ext k
  simp only [coeff_mul_monomial, coeff_monomial]
  split_ifs with h₁ h₂ h₃ h₃ h₂ <;> try rfl
  · rw [← h₂, tsub_add_cancel_of_le h₁] at h₃
    exact (h₃ rfl).elim
  · rw [h₃, add_tsub_cancel_right] at h₂
    exact (h₂ rfl).elim
  · exact zero_mul b
  · rw [h₂] at h₁
    exact (h₁ <| le_add_left le_rfl).elim
theorem monomial_mul_monomial (m n : σ →₀ ℕ) (a b : R) :
    monomial R m a * monomial R n b = monomial R (m + n) (a * b) := by
  classical
  

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

Next tactic:
ext k
theorem monomial_mul_monomial (m n : σ →₀ ℕ) (a b : R) :
    monomial R m a * monomial R n b = monomial R (m + n) (a * b) := by
  classical
  ext k
  

Proof state:
case h
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
⊢ (coeff R k) ((monomial R m) a * (monomial R n) b) = (coeff R k) ((monomial R (m + n)) (a * b))

Next tactic:
simp only [coeff_mul_monomial, coeff_monomial]
theorem monomial_mul_monomial (m n : σ →₀ ℕ) (a b : R) :
    monomial R m a * monomial R n b = monomial R (m + n) (a * b) := by
  classical
  ext k
  simp only [coeff_mul_monomial, coeff_monomial]
  

Proof state:
case h
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
⊢ (if n ≤ k then (if k - n = m then a else 0) * b else 0) = if k = m + n then a * b else 0

Next tactic:
split_ifs with h₁ h₂ h₃ h₃ h₂
theorem monomial_mul_monomial (m n : σ →₀ ℕ) (a b : R) :
    monomial R m a * monomial R n b = monomial R (m + n) (a * b) := by
  classical
  ext k
  simp only [coeff_mul_monomial, coeff_monomial]
  split_ifs with h₁ h₂ h₃ h₃ h₂ <;> 

Proof state:
case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : n ≤ k
h₂ : k - n = m
h₃ : k = m + n
⊢ a * b = a * b

Next tactic:
try rfl
theorem monomial_mul_monomial (m n : σ →₀ ℕ) (a b : R) :
    monomial R m a * monomial R n b = monomial R (m + n) (a * b) := by
  classical
  ext k
  simp only [coeff_mul_monomial, coeff_monomial]
  split_ifs with h₁ h₂ h₃ h₃ h₂ <;> try 

Proof state:
case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : n ≤ k
h₂ : k - n = m
h₃ : k = m + n
⊢ a * b = a * b

Next tactic:
rfl
theorem monomial_mul_monomial (m n : σ →₀ ℕ) (a b : R) :
    monomial R m a * monomial R n b = monomial R (m + n) (a * b) := by
  classical
  ext k
  simp only [coeff_mul_monomial, coeff_monomial]
  split_ifs with h₁ h₂ h₃ h₃ h₂ <;> 

Proof state:
case neg
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : n ≤ k
h₂ : k - n = m
h₃ : ¬k = m + n
⊢ a * b = 0

Next tactic:
try rfl
theorem monomial_mul_monomial (m n : σ →₀ ℕ) (a b : R) :
    monomial R m a * monomial R n b = monomial R (m + n) (a * b) := by
  classical
  ext k
  simp only [coeff_mul_monomial, coeff_monomial]
  split_ifs with h₁ h₂ h₃ h₃ h₂ <;> try 

Proof state:
case neg
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : n ≤ k
h₂ : k - n = m
h₃ : ¬k = m + n
⊢ a * b = 0

Next tactic:
rfl
theorem monomial_mul_monomial (m n : σ →₀ ℕ) (a b : R) :
    monomial R m a * monomial R n b = monomial R (m + n) (a * b) := by
  classical
  ext k
  simp only [coeff_mul_monomial, coeff_monomial]
  split_ifs with h₁ h₂ h₃ h₃ h₂ <;> 

Proof state:
case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : n ≤ k
h₂ : ¬k - n = m
h₃ : k = m + n
⊢ 0 * b = a * b

Next tactic:
try rfl
theorem monomial_mul_monomial (m n : σ →₀ ℕ) (a b : R) :
    monomial R m a * monomial R n b = monomial R (m + n) (a * b) := by
  classical
  ext k
  simp only [coeff_mul_monomial, coeff_monomial]
  split_ifs with h₁ h₂ h₃ h₃ h₂ <;> try 

Proof state:
case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : n ≤ k
h₂ : ¬k - n = m
h₃ : k = m + n
⊢ 0 * b = a * b

Next tactic:
rfl
theorem monomial_mul_monomial (m n : σ →₀ ℕ) (a b : R) :
    monomial R m a * monomial R n b = monomial R (m + n) (a * b) := by
  classical
  ext k
  simp only [coeff_mul_monomial, coeff_monomial]
  split_ifs with h₁ h₂ h₃ h₃ h₂ <;> 

Proof state:
case neg
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : n ≤ k
h₂ : ¬k - n = m
h₃ : ¬k = m + n
⊢ 0 * b = 0

Next tactic:
try rfl
theorem monomial_mul_monomial (m n : σ →₀ ℕ) (a b : R) :
    monomial R m a * monomial R n b = monomial R (m + n) (a * b) := by
  classical
  ext k
  simp only [coeff_mul_monomial, coeff_monomial]
  split_ifs with h₁ h₂ h₃ h₃ h₂ <;> try 

Proof state:
case neg
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : n ≤ k
h₂ : ¬k - n = m
h₃ : ¬k = m + n
⊢ 0 * b = 0

Next tactic:
rfl
theorem monomial_mul_monomial (m n : σ →₀ ℕ) (a b : R) :
    monomial R m a * monomial R n b = monomial R (m + n) (a * b) := by
  classical
  ext k
  simp only [coeff_mul_monomial, coeff_monomial]
  split_ifs with h₁ h₂ h₃ h₃ h₂ <;> 

Proof state:
case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : ¬n ≤ k
h₂ : k = m + n
⊢ 0 = a * b

Next tactic:
try rfl
theorem monomial_mul_monomial (m n : σ →₀ ℕ) (a b : R) :
    monomial R m a * monomial R n b = monomial R (m + n) (a * b) := by
  classical
  ext k
  simp only [coeff_mul_monomial, coeff_monomial]
  split_ifs with h₁ h₂ h₃ h₃ h₂ <;> try 

Proof state:
case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : ¬n ≤ k
h₂ : k = m + n
⊢ 0 = a * b

Next tactic:
rfl
theorem monomial_mul_monomial (m n : σ →₀ ℕ) (a b : R) :
    monomial R m a * monomial R n b = monomial R (m + n) (a * b) := by
  classical
  ext k
  simp only [coeff_mul_monomial, coeff_monomial]
  split_ifs with h₁ h₂ h₃ h₃ h₂ <;> 

Proof state:
case neg
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : ¬n ≤ k
h₂ : ¬k = m + n
⊢ 0 = 0

Next tactic:
try rfl
theorem monomial_mul_monomial (m n : σ →₀ ℕ) (a b : R) :
    monomial R m a * monomial R n b = monomial R (m + n) (a * b) := by
  classical
  ext k
  simp only [coeff_mul_monomial, coeff_monomial]
  split_ifs with h₁ h₂ h₃ h₃ h₂ <;> try 

Proof state:
case neg
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : ¬n ≤ k
h₂ : ¬k = m + n
⊢ 0 = 0

Next tactic:
rfl
theorem monomial_mul_monomial (m n : σ →₀ ℕ) (a b : R) :
    monomial R m a * monomial R n b = monomial R (m + n) (a * b) := by
  classical
  ext k
  simp only [coeff_mul_monomial, coeff_monomial]
  split_ifs with h₁ h₂ h₃ h₃ h₂ <;> try rfl
  · 

Proof state:
case neg
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
a b : R
k : σ →₀ ℕ
h₁ : n ≤ k
h₂ : k - n = m
h₃ : ¬k = m + n
⊢ a * b = 0

Next tactic:
rw [← h₂, tsub_add_cancel_of_le h₁] at h₃