l3lab/ntp-mathlib huggingface dataset:
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    intros
    first
      | rfl
      | ext; simp only [add_apply, mul_apply]
        first
          | rw [mul_add]
          | 

Proof state:
case refine'_5.a
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
a✝ b✝ c✝ : Poly α
x✝ : α → ℕ
⊢ a✝ x✝ * b✝ x✝ * c✝ x✝ = a✝ x✝ * (b✝ x✝ * c✝ x✝)

Next tactic:
rw [add_mul]
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    intros
    first
      | rfl
      | ext; simp only [add_apply, mul_apply]
        first
          | rw [mul_add]
          | rw [add_mul]
          | 

Proof state:
case refine'_5.a
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
a✝ b✝ c✝ : Poly α
x✝ : α → ℕ
⊢ a✝ x✝ * b✝ x✝ * c✝ x✝ = a✝ x✝ * (b✝ x✝ * c✝ x✝)

Next tactic:
rw [zero_apply, zero_mul]
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    intros
    first
      | rfl
      | ext; simp only [add_apply, mul_apply]
        first
          | rw [mul_add]
          | rw [add_mul]
          | rw [zero_apply, zero_mul]
          | 

Proof state:
case refine'_5.a
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
a✝ b✝ c✝ : Poly α
x✝ : α → ℕ
⊢ a✝ x✝ * b✝ x✝ * c✝ x✝ = a✝ x✝ * (b✝ x✝ * c✝ x✝)

Next tactic:
rw [zero_apply, mul_zero]
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    intros
    first
      | rfl
      | ext; simp only [add_apply, mul_apply]
        first
          | rw [mul_add]
          | rw [add_mul]
          | rw [zero_apply, zero_mul]
          | rw [zero_apply, mul_zero]
          | 

Proof state:
case refine'_5.a
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
a✝ b✝ c✝ : Poly α
x✝ : α → ℕ
⊢ a✝ x✝ * b✝ x✝ * c✝ x✝ = a✝ x✝ * (b✝ x✝ * c✝ x✝)

Next tactic:
rw [← mul_assoc]
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    

Proof state:
case refine'_6
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
⊢ ∀ (a : Poly α), 1 * a = a

Next tactic:
intros
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    intros
    

Proof state:
case refine'_6
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
a✝ : Poly α
⊢ 1 * a✝ = a✝

Next tactic:
first
      | rfl
      | ext; simp only [add_apply, mul_apply]
        first
          | rw [mul_add]
          | rw [add_mul]
          | rw [zero_apply, zero_mul]
          | rw [zero_apply, mul_zero]
          | rw [← mul_assoc]
          | rw [one_apply, one_mul]
          | rw [one_apply, mul_one]
          | rw [mul_comm]
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    intros
    first
      | 

Proof state:
case refine'_6
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
a✝ : Poly α
⊢ 1 * a✝ = a✝

Next tactic:
rfl
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    intros
    first
      | rfl
      | 

Proof state:
case refine'_6
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
a✝ : Poly α
⊢ 1 * a✝ = a✝

Next tactic:
ext
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    intros
    first
      | rfl
      | ext; 

Proof state:
case refine'_6.a
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
a✝ : Poly α
x✝ : α → ℕ
⊢ (1 * a✝) x✝ = a✝ x✝

Next tactic:
simp only [add_apply, mul_apply]
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    intros
    first
      | rfl
      | ext; simp only [add_apply, mul_apply]
        

Proof state:
case refine'_6.a
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
a✝ : Poly α
x✝ : α → ℕ
⊢ 1 x✝ * a✝ x✝ = a✝ x✝

Next tactic:
first
          | rw [mul_add]
          | rw [add_mul]
          | rw [zero_apply, zero_mul]
          | rw [zero_apply, mul_zero]
          | rw [← mul_assoc]
          | rw [one_apply, one_mul]
          | rw [one_apply, mul_one]
          | rw [mul_comm]
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    intros
    first
      | rfl
      | ext; simp only [add_apply, mul_apply]
        first
          | 

Proof state:
case refine'_6.a
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
a✝ : Poly α
x✝ : α → ℕ
⊢ 1 x✝ * a✝ x✝ = a✝ x✝

Next tactic:
rw [mul_add]
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    intros
    first
      | rfl
      | ext; simp only [add_apply, mul_apply]
        first
          | rw [mul_add]
          | 

Proof state:
case refine'_6.a
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
a✝ : Poly α
x✝ : α → ℕ
⊢ 1 x✝ * a✝ x✝ = a✝ x✝

Next tactic:
rw [add_mul]
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    intros
    first
      | rfl
      | ext; simp only [add_apply, mul_apply]
        first
          | rw [mul_add]
          | rw [add_mul]
          | 

Proof state:
case refine'_6.a
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
a✝ : Poly α
x✝ : α → ℕ
⊢ 1 x✝ * a✝ x✝ = a✝ x✝

Next tactic:
rw [zero_apply, zero_mul]
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    intros
    first
      | rfl
      | ext; simp only [add_apply, mul_apply]
        first
          | rw [mul_add]
          | rw [add_mul]
          | rw [zero_apply, zero_mul]
          | 

Proof state:
case refine'_6.a
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
a✝ : Poly α
x✝ : α → ℕ
⊢ 1 x✝ * a✝ x✝ = a✝ x✝

Next tactic:
rw [zero_apply, mul_zero]
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    intros
    first
      | rfl
      | ext; simp only [add_apply, mul_apply]
        first
          | rw [mul_add]
          | rw [add_mul]
          | rw [zero_apply, zero_mul]
          | rw [zero_apply, mul_zero]
          | 

Proof state:
case refine'_6.a
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
a✝ : Poly α
x✝ : α → ℕ
⊢ 1 x✝ * a✝ x✝ = a✝ x✝

Next tactic:
rw [← mul_assoc]
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    intros
    first
      | rfl
      | ext; simp only [add_apply, mul_apply]
        first
          | rw [mul_add]
          | rw [add_mul]
          | rw [zero_apply, zero_mul]
          | rw [zero_apply, mul_zero]
          | rw [← mul_assoc]
          | 

Proof state:
case refine'_6.a
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
a✝ : Poly α
x✝ : α → ℕ
⊢ 1 x✝ * a✝ x✝ = a✝ x✝

Next tactic:
rw [one_apply, one_mul]
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    

Proof state:
case refine'_7
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
⊢ ∀ (a : Poly α), a * 1 = a

Next tactic:
intros
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    intros
    

Proof state:
case refine'_7
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
a✝ : Poly α
⊢ a✝ * 1 = a✝

Next tactic:
first
      | rfl
      | ext; simp only [add_apply, mul_apply]
        first
          | rw [mul_add]
          | rw [add_mul]
          | rw [zero_apply, zero_mul]
          | rw [zero_apply, mul_zero]
          | rw [← mul_assoc]
          | rw [one_apply, one_mul]
          | rw [one_apply, mul_one]
          | rw [mul_comm]
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    intros
    first
      | 

Proof state:
case refine'_7
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
a✝ : Poly α
⊢ a✝ * 1 = a✝

Next tactic:
rfl
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    intros
    first
      | rfl
      | 

Proof state:
case refine'_7
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
a✝ : Poly α
⊢ a✝ * 1 = a✝

Next tactic:
ext
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    intros
    first
      | rfl
      | ext; 

Proof state:
case refine'_7.a
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
a✝ : Poly α
x✝ : α → ℕ
⊢ (a✝ * 1) x✝ = a✝ x✝

Next tactic:
simp only [add_apply, mul_apply]
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    intros
    first
      | rfl
      | ext; simp only [add_apply, mul_apply]
        

Proof state:
case refine'_7.a
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
a✝ : Poly α
x✝ : α → ℕ
⊢ a✝ x✝ * 1 x✝ = a✝ x✝

Next tactic:
first
          | rw [mul_add]
          | rw [add_mul]
          | rw [zero_apply, zero_mul]
          | rw [zero_apply, mul_zero]
          | rw [← mul_assoc]
          | rw [one_apply, one_mul]
          | rw [one_apply, mul_one]
          | rw [mul_comm]
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    intros
    first
      | rfl
      | ext; simp only [add_apply, mul_apply]
        first
          | 

Proof state:
case refine'_7.a
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
a✝ : Poly α
x✝ : α → ℕ
⊢ a✝ x✝ * 1 x✝ = a✝ x✝

Next tactic:
rw [mul_add]
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    intros
    first
      | rfl
      | ext; simp only [add_apply, mul_apply]
        first
          | rw [mul_add]
          | 

Proof state:
case refine'_7.a
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
a✝ : Poly α
x✝ : α → ℕ
⊢ a✝ x✝ * 1 x✝ = a✝ x✝

Next tactic:
rw [add_mul]
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    intros
    first
      | rfl
      | ext; simp only [add_apply, mul_apply]
        first
          | rw [mul_add]
          | rw [add_mul]
          | 

Proof state:
case refine'_7.a
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
a✝ : Poly α
x✝ : α → ℕ
⊢ a✝ x✝ * 1 x✝ = a✝ x✝

Next tactic:
rw [zero_apply, zero_mul]
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    intros
    first
      | rfl
      | ext; simp only [add_apply, mul_apply]
        first
          | rw [mul_add]
          | rw [add_mul]
          | rw [zero_apply, zero_mul]
          | 

Proof state:
case refine'_7.a
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
a✝ : Poly α
x✝ : α → ℕ
⊢ a✝ x✝ * 1 x✝ = a✝ x✝

Next tactic:
rw [zero_apply, mul_zero]
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    intros
    first
      | rfl
      | ext; simp only [add_apply, mul_apply]
        first
          | rw [mul_add]
          | rw [add_mul]
          | rw [zero_apply, zero_mul]
          | rw [zero_apply, mul_zero]
          | 

Proof state:
case refine'_7.a
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
a✝ : Poly α
x✝ : α → ℕ
⊢ a✝ x✝ * 1 x✝ = a✝ x✝

Next tactic:
rw [← mul_assoc]
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    intros
    first
      | rfl
      | ext; simp only [add_apply, mul_apply]
        first
          | rw [mul_add]
          | rw [add_mul]
          | rw [zero_apply, zero_mul]
          | rw [zero_apply, mul_zero]
          | rw [← mul_assoc]
          | 

Proof state:
case refine'_7.a
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
a✝ : Poly α
x✝ : α → ℕ
⊢ a✝ x✝ * 1 x✝ = a✝ x✝

Next tactic:
rw [one_apply, one_mul]
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    intros
    first
      | rfl
      | ext; simp only [add_apply, mul_apply]
        first
          | rw [mul_add]
          | rw [add_mul]
          | rw [zero_apply, zero_mul]
          | rw [zero_apply, mul_zero]
          | rw [← mul_assoc]
          | rw [one_apply, one_mul]
          | 

Proof state:
case refine'_7.a
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
a✝ : Poly α
x✝ : α → ℕ
⊢ a✝ x✝ * 1 x✝ = a✝ x✝

Next tactic:
rw [one_apply, mul_one]
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    

Proof state:
case refine'_8
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
⊢ ∀ (x : Poly α), npowRec 0 x = 1

Next tactic:
intros
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    intros
    

Proof state:
case refine'_8
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
x✝ : Poly α
⊢ npowRec 0 x✝ = 1

Next tactic:
first
      | rfl
      | ext; simp only [add_apply, mul_apply]
        first
          | rw [mul_add]
          | rw [add_mul]
          | rw [zero_apply, zero_mul]
          | rw [zero_apply, mul_zero]
          | rw [← mul_assoc]
          | rw [one_apply, one_mul]
          | rw [one_apply, mul_one]
          | rw [mul_comm]
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    intros
    first
      | 

Proof state:
case refine'_8
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
x✝ : Poly α
⊢ npowRec 0 x✝ = 1

Next tactic:
rfl
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    

Proof state:
case refine'_9
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
⊢ ∀ (n : ℕ) (x : Poly α), npowRec (n + 1) x = x * npowRec n x

Next tactic:
intros
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    intros
    

Proof state:
case refine'_9
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
n✝ : ℕ
x✝ : Poly α
⊢ npowRec (n✝ + 1) x✝ = x✝ * npowRec n✝ x✝

Next tactic:
first
      | rfl
      | ext; simp only [add_apply, mul_apply]
        first
          | rw [mul_add]
          | rw [add_mul]
          | rw [zero_apply, zero_mul]
          | rw [zero_apply, mul_zero]
          | rw [← mul_assoc]
          | rw [one_apply, one_mul]
          | rw [one_apply, mul_one]
          | rw [mul_comm]
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    intros
    first
      | 

Proof state:
case refine'_9
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
n✝ : ℕ
x✝ : Poly α
⊢ npowRec (n✝ + 1) x✝ = x✝ * npowRec n✝ x✝

Next tactic:
rfl
instance : CommRing (Poly α) := by
  refine' { (inferInstance : AddCommGroup (Poly α)),
            (inferInstance : AddGroupWithOne (Poly α)) with
              mul := (· * ·)
              npow := @npowRec _ ⟨(1 : Poly α)⟩ ⟨(· * ·)⟩
              .. }
  all_goals
    

Proof state:
case refine'_10
α : Type u_1
β : Type u_2
γ : Type u_3
src✝¹ : AddCommGroup (Poly α) := inferInstance
src✝ : AddGroupWithOne (Poly α) := inferInstance
⊢ ∀ (a b : Poly α), a * b = b * a

Next tactic:
intros