l3lab/ntp-mathlib huggingface dataset:
instance addCommGroup : AddCommGroup ℂ :=
  { zero := (0 : ℂ)
    add := (· + ·)
    neg := Neg.neg
    sub := Sub.sub
    nsmul := fun n z => n • z
    zsmul := fun n z => n • z
    zsmul_zero' := by intros; ext <;> simp [smul_re, smul_im]
    nsmul_zero := by intros; ext <;> simp [smul_re, smul_im]
    nsmul_succ := by
      intros; ext <;> simp [AddMonoid.nsmul_succ, add_mul, add_comm,
        smul_re, smul_im]
    zsmul_succ' := by
      intros; ext <;> simp [SubNegMonoid.zsmul_succ', add_mul, add_comm,
        smul_re, smul_im]
    zsmul_neg' := by
      intros; ext <;> simp [zsmul_neg', add_mul, smul_re, smul_im]
    add_assoc := by intros; ext <;> simp [add_assoc]
    zero_add := by intros; ext <;> simp
    add_zero := by intros; ext <;> simp
    add_comm := by intros; 

Proof state:
a✝ b✝ : ℂ
⊢ a✝ + b✝ = b✝ + a✝

Next tactic:
ext
instance addCommGroup : AddCommGroup ℂ :=
  { zero := (0 : ℂ)
    add := (· + ·)
    neg := Neg.neg
    sub := Sub.sub
    nsmul := fun n z => n • z
    zsmul := fun n z => n • z
    zsmul_zero' := by intros; ext <;> simp [smul_re, smul_im]
    nsmul_zero := by intros; ext <;> simp [smul_re, smul_im]
    nsmul_succ := by
      intros; ext <;> simp [AddMonoid.nsmul_succ, add_mul, add_comm,
        smul_re, smul_im]
    zsmul_succ' := by
      intros; ext <;> simp [SubNegMonoid.zsmul_succ', add_mul, add_comm,
        smul_re, smul_im]
    zsmul_neg' := by
      intros; ext <;> simp [zsmul_neg', add_mul, smul_re, smul_im]
    add_assoc := by intros; ext <;> simp [add_assoc]
    zero_add := by intros; ext <;> simp
    add_zero := by intros; ext <;> simp
    add_comm := by intros; ext <;> 

Proof state:
case a
a✝ b✝ : ℂ
⊢ (a✝ + b✝).re = (b✝ + a✝).re

Next tactic:
simp [add_comm]
instance addCommGroup : AddCommGroup ℂ :=
  { zero := (0 : ℂ)
    add := (· + ·)
    neg := Neg.neg
    sub := Sub.sub
    nsmul := fun n z => n • z
    zsmul := fun n z => n • z
    zsmul_zero' := by intros; ext <;> simp [smul_re, smul_im]
    nsmul_zero := by intros; ext <;> simp [smul_re, smul_im]
    nsmul_succ := by
      intros; ext <;> simp [AddMonoid.nsmul_succ, add_mul, add_comm,
        smul_re, smul_im]
    zsmul_succ' := by
      intros; ext <;> simp [SubNegMonoid.zsmul_succ', add_mul, add_comm,
        smul_re, smul_im]
    zsmul_neg' := by
      intros; ext <;> simp [zsmul_neg', add_mul, smul_re, smul_im]
    add_assoc := by intros; ext <;> simp [add_assoc]
    zero_add := by intros; ext <;> simp
    add_zero := by intros; ext <;> simp
    add_comm := by intros; ext <;> 

Proof state:
case a
a✝ b✝ : ℂ
⊢ (a✝ + b✝).im = (b✝ + a✝).im

Next tactic:
simp [add_comm]
instance Complex.addGroupWithOne : AddGroupWithOne ℂ :=
  { Complex.addCommGroup with
    natCast := fun n => ⟨n, 0⟩
    natCast_zero := by
      

Proof state:
src✝ : AddCommGroup ℂ := addCommGroup
⊢ NatCast.natCast 0 = 0

Next tactic:
ext
instance Complex.addGroupWithOne : AddGroupWithOne ℂ :=
  { Complex.addCommGroup with
    natCast := fun n => ⟨n, 0⟩
    natCast_zero := by
      ext <;> 

Proof state:
case a
src✝ : AddCommGroup ℂ := addCommGroup
⊢ (NatCast.natCast 0).re = 0.re

Next tactic:
simp [Nat.cast, AddMonoidWithOne.natCast_zero]
instance Complex.addGroupWithOne : AddGroupWithOne ℂ :=
  { Complex.addCommGroup with
    natCast := fun n => ⟨n, 0⟩
    natCast_zero := by
      ext <;> 

Proof state:
case a
src✝ : AddCommGroup ℂ := addCommGroup
⊢ (NatCast.natCast 0).im = 0.im

Next tactic:
simp [Nat.cast, AddMonoidWithOne.natCast_zero]
instance Complex.addGroupWithOne : AddGroupWithOne ℂ :=
  { Complex.addCommGroup with
    natCast := fun n => ⟨n, 0⟩
    natCast_zero := by
      ext <;> simp [Nat.cast, AddMonoidWithOne.natCast_zero]
    natCast_succ := fun _ => by 

Proof state:
src✝ : AddCommGroup ℂ := addCommGroup
x✝ : ℕ
⊢ NatCast.natCast (x✝ + 1) = NatCast.natCast x✝ + 1

Next tactic:
ext
instance Complex.addGroupWithOne : AddGroupWithOne ℂ :=
  { Complex.addCommGroup with
    natCast := fun n => ⟨n, 0⟩
    natCast_zero := by
      ext <;> simp [Nat.cast, AddMonoidWithOne.natCast_zero]
    natCast_succ := fun _ => by ext <;> 

Proof state:
case a
src✝ : AddCommGroup ℂ := addCommGroup
x✝ : ℕ
⊢ (NatCast.natCast (x✝ + 1)).re = (NatCast.natCast x✝ + 1).re

Next tactic:
simp [Nat.cast, AddMonoidWithOne.natCast_succ]
instance Complex.addGroupWithOne : AddGroupWithOne ℂ :=
  { Complex.addCommGroup with
    natCast := fun n => ⟨n, 0⟩
    natCast_zero := by
      ext <;> simp [Nat.cast, AddMonoidWithOne.natCast_zero]
    natCast_succ := fun _ => by ext <;> 

Proof state:
case a
src✝ : AddCommGroup ℂ := addCommGroup
x✝ : ℕ
⊢ (NatCast.natCast (x✝ + 1)).im = (NatCast.natCast x✝ + 1).im

Next tactic:
simp [Nat.cast, AddMonoidWithOne.natCast_succ]
instance Complex.addGroupWithOne : AddGroupWithOne ℂ :=
  { Complex.addCommGroup with
    natCast := fun n => ⟨n, 0⟩
    natCast_zero := by
      ext <;> simp [Nat.cast, AddMonoidWithOne.natCast_zero]
    natCast_succ := fun _ => by ext <;> simp [Nat.cast, AddMonoidWithOne.natCast_succ]
    intCast := fun n => ⟨n, 0⟩
    intCast_ofNat := fun _ => by 

Proof state:
src✝ : AddCommGroup ℂ := addCommGroup
x✝ : ℕ
⊢ IntCast.intCast ↑x✝ = ↑x✝

Next tactic:
ext
instance Complex.addGroupWithOne : AddGroupWithOne ℂ :=
  { Complex.addCommGroup with
    natCast := fun n => ⟨n, 0⟩
    natCast_zero := by
      ext <;> simp [Nat.cast, AddMonoidWithOne.natCast_zero]
    natCast_succ := fun _ => by ext <;> simp [Nat.cast, AddMonoidWithOne.natCast_succ]
    intCast := fun n => ⟨n, 0⟩
    intCast_ofNat := fun _ => by ext <;> 

Proof state:
case a
src✝ : AddCommGroup ℂ := addCommGroup
x✝ : ℕ
⊢ (IntCast.intCast ↑x✝).re = (↑x✝).re

Next tactic:
rfl
instance Complex.addGroupWithOne : AddGroupWithOne ℂ :=
  { Complex.addCommGroup with
    natCast := fun n => ⟨n, 0⟩
    natCast_zero := by
      ext <;> simp [Nat.cast, AddMonoidWithOne.natCast_zero]
    natCast_succ := fun _ => by ext <;> simp [Nat.cast, AddMonoidWithOne.natCast_succ]
    intCast := fun n => ⟨n, 0⟩
    intCast_ofNat := fun _ => by ext <;> 

Proof state:
case a
src✝ : AddCommGroup ℂ := addCommGroup
x✝ : ℕ
⊢ (IntCast.intCast ↑x✝).im = (↑x✝).im

Next tactic:
rfl
instance Complex.addGroupWithOne : AddGroupWithOne ℂ :=
  { Complex.addCommGroup with
    natCast := fun n => ⟨n, 0⟩
    natCast_zero := by
      ext <;> simp [Nat.cast, AddMonoidWithOne.natCast_zero]
    natCast_succ := fun _ => by ext <;> simp [Nat.cast, AddMonoidWithOne.natCast_succ]
    intCast := fun n => ⟨n, 0⟩
    intCast_ofNat := fun _ => by ext <;> rfl
    intCast_negSucc := fun n => by
      

Proof state:
src✝ : AddCommGroup ℂ := addCommGroup
n : ℕ
⊢ IntCast.intCast (Int.negSucc n) = -↑(n + 1)

Next tactic:
ext
instance Complex.addGroupWithOne : AddGroupWithOne ℂ :=
  { Complex.addCommGroup with
    natCast := fun n => ⟨n, 0⟩
    natCast_zero := by
      ext <;> simp [Nat.cast, AddMonoidWithOne.natCast_zero]
    natCast_succ := fun _ => by ext <;> simp [Nat.cast, AddMonoidWithOne.natCast_succ]
    intCast := fun n => ⟨n, 0⟩
    intCast_ofNat := fun _ => by ext <;> rfl
    intCast_negSucc := fun n => by
      ext
      · 

Proof state:
case a
src✝ : AddCommGroup ℂ := addCommGroup
n : ℕ
⊢ (IntCast.intCast (Int.negSucc n)).re = (-↑(n + 1)).re

Next tactic:
simp [AddGroupWithOne.intCast_negSucc]
instance Complex.addGroupWithOne : AddGroupWithOne ℂ :=
  { Complex.addCommGroup with
    natCast := fun n => ⟨n, 0⟩
    natCast_zero := by
      ext <;> simp [Nat.cast, AddMonoidWithOne.natCast_zero]
    natCast_succ := fun _ => by ext <;> simp [Nat.cast, AddMonoidWithOne.natCast_succ]
    intCast := fun n => ⟨n, 0⟩
    intCast_ofNat := fun _ => by ext <;> rfl
    intCast_negSucc := fun n => by
      ext
      · simp [AddGroupWithOne.intCast_negSucc]
        

Proof state:
case a
src✝ : AddCommGroup ℂ := addCommGroup
n : ℕ
⊢ -1 + -↑n = -(↑(n + 1)).re

Next tactic:
show -(1: ℝ) + (-n) = -(↑(n + 1))
instance Complex.addGroupWithOne : AddGroupWithOne ℂ :=
  { Complex.addCommGroup with
    natCast := fun n => ⟨n, 0⟩
    natCast_zero := by
      ext <;> simp [Nat.cast, AddMonoidWithOne.natCast_zero]
    natCast_succ := fun _ => by ext <;> simp [Nat.cast, AddMonoidWithOne.natCast_succ]
    intCast := fun n => ⟨n, 0⟩
    intCast_ofNat := fun _ => by ext <;> rfl
    intCast_negSucc := fun n => by
      ext
      · simp [AddGroupWithOne.intCast_negSucc]
        show -(1: ℝ) + (-n) = -(↑(n + 1))
        

Proof state:
case a
src✝ : AddCommGroup ℂ := addCommGroup
n : ℕ
⊢ -1 + -↑n = -↑(n + 1)

Next tactic:
simp [Nat.cast_add, add_comm]
instance Complex.addGroupWithOne : AddGroupWithOne ℂ :=
  { Complex.addCommGroup with
    natCast := fun n => ⟨n, 0⟩
    natCast_zero := by
      ext <;> simp [Nat.cast, AddMonoidWithOne.natCast_zero]
    natCast_succ := fun _ => by ext <;> simp [Nat.cast, AddMonoidWithOne.natCast_succ]
    intCast := fun n => ⟨n, 0⟩
    intCast_ofNat := fun _ => by ext <;> rfl
    intCast_negSucc := fun n => by
      ext
      · simp [AddGroupWithOne.intCast_negSucc]
        show -(1: ℝ) + (-n) = -(↑(n + 1))
        simp [Nat.cast_add, add_comm]
      · 

Proof state:
case a
src✝ : AddCommGroup ℂ := addCommGroup
n : ℕ
⊢ (IntCast.intCast (Int.negSucc n)).im = (-↑(n + 1)).im

Next tactic:
simp [AddGroupWithOne.intCast_negSucc]
instance Complex.addGroupWithOne : AddGroupWithOne ℂ :=
  { Complex.addCommGroup with
    natCast := fun n => ⟨n, 0⟩
    natCast_zero := by
      ext <;> simp [Nat.cast, AddMonoidWithOne.natCast_zero]
    natCast_succ := fun _ => by ext <;> simp [Nat.cast, AddMonoidWithOne.natCast_succ]
    intCast := fun n => ⟨n, 0⟩
    intCast_ofNat := fun _ => by ext <;> rfl
    intCast_negSucc := fun n => by
      ext
      · simp [AddGroupWithOne.intCast_negSucc]
        show -(1: ℝ) + (-n) = -(↑(n + 1))
        simp [Nat.cast_add, add_comm]
      · simp [AddGroupWithOne.intCast_negSucc]
        

Proof state:
case a
src✝ : AddCommGroup ℂ := addCommGroup
n : ℕ
⊢ (↑(n + 1)).im = 0

Next tactic:
show im ⟨n, 0⟩ = 0
instance Complex.addGroupWithOne : AddGroupWithOne ℂ :=
  { Complex.addCommGroup with
    natCast := fun n => ⟨n, 0⟩
    natCast_zero := by
      ext <;> simp [Nat.cast, AddMonoidWithOne.natCast_zero]
    natCast_succ := fun _ => by ext <;> simp [Nat.cast, AddMonoidWithOne.natCast_succ]
    intCast := fun n => ⟨n, 0⟩
    intCast_ofNat := fun _ => by ext <;> rfl
    intCast_negSucc := fun n => by
      ext
      · simp [AddGroupWithOne.intCast_negSucc]
        show -(1: ℝ) + (-n) = -(↑(n + 1))
        simp [Nat.cast_add, add_comm]
      · simp [AddGroupWithOne.intCast_negSucc]
        show im ⟨n, 0⟩ = 0
        

Proof state:
case a
src✝ : AddCommGroup ℂ := addCommGroup
n : ℕ
⊢ { re := ↑n, im := 0 }.im = 0

Next tactic:
rfl
instance commRing : CommRing ℂ :=
  { Complex.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec _ ⟨(1 : ℂ)⟩ ⟨(· * ·)⟩
    add_comm := by 

Proof state:
src✝ : AddGroupWithOne ℂ := Complex.addGroupWithOne
⊢ ∀ (a b : ℂ), a + b = b + a

Next tactic:
intros
instance commRing : CommRing ℂ :=
  { Complex.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec _ ⟨(1 : ℂ)⟩ ⟨(· * ·)⟩
    add_comm := by intros; 

Proof state:
src✝ : AddGroupWithOne ℂ := Complex.addGroupWithOne
a✝ b✝ : ℂ
⊢ a✝ + b✝ = b✝ + a✝

Next tactic:
ext
instance commRing : CommRing ℂ :=
  { Complex.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec _ ⟨(1 : ℂ)⟩ ⟨(· * ·)⟩
    add_comm := by intros; ext <;> 

Proof state:
case a
src✝ : AddGroupWithOne ℂ := Complex.addGroupWithOne
a✝ b✝ : ℂ
⊢ (a✝ + b✝).re = (b✝ + a✝).re

Next tactic:
simp [add_comm]
instance commRing : CommRing ℂ :=
  { Complex.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec _ ⟨(1 : ℂ)⟩ ⟨(· * ·)⟩
    add_comm := by intros; ext <;> 

Proof state:
case a
src✝ : AddGroupWithOne ℂ := Complex.addGroupWithOne
a✝ b✝ : ℂ
⊢ (a✝ + b✝).im = (b✝ + a✝).im

Next tactic:
simp [add_comm]
instance commRing : CommRing ℂ :=
  { Complex.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec _ ⟨(1 : ℂ)⟩ ⟨(· * ·)⟩
    add_comm := by intros; ext <;> simp [add_comm]
    left_distrib := by
      

Proof state:
src✝ : AddGroupWithOne ℂ := Complex.addGroupWithOne
⊢ ∀ (a b c : ℂ), a * (b + c) = a * b + a * c

Next tactic:
intros
instance commRing : CommRing ℂ :=
  { Complex.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec _ ⟨(1 : ℂ)⟩ ⟨(· * ·)⟩
    add_comm := by intros; ext <;> simp [add_comm]
    left_distrib := by
      intros; 

Proof state:
src✝ : AddGroupWithOne ℂ := Complex.addGroupWithOne
a✝ b✝ c✝ : ℂ
⊢ a✝ * (b✝ + c✝) = a✝ * b✝ + a✝ * c✝

Next tactic:
ext
instance commRing : CommRing ℂ :=
  { Complex.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec _ ⟨(1 : ℂ)⟩ ⟨(· * ·)⟩
    add_comm := by intros; ext <;> simp [add_comm]
    left_distrib := by
      intros; ext <;> 

Proof state:
case a
src✝ : AddGroupWithOne ℂ := Complex.addGroupWithOne
a✝ b✝ c✝ : ℂ
⊢ (a✝ * (b✝ + c✝)).re = (a✝ * b✝ + a✝ * c✝).re

Next tactic:
simp [mul_re, mul_im]
instance commRing : CommRing ℂ :=
  { Complex.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec _ ⟨(1 : ℂ)⟩ ⟨(· * ·)⟩
    add_comm := by intros; ext <;> simp [add_comm]
    left_distrib := by
      intros; ext <;> 

Proof state:
case a
src✝ : AddGroupWithOne ℂ := Complex.addGroupWithOne
a✝ b✝ c✝ : ℂ
⊢ (a✝ * (b✝ + c✝)).im = (a✝ * b✝ + a✝ * c✝).im

Next tactic:
simp [mul_re, mul_im]
instance commRing : CommRing ℂ :=
  { Complex.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec _ ⟨(1 : ℂ)⟩ ⟨(· * ·)⟩
    add_comm := by intros; ext <;> simp [add_comm]
    left_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> 

Proof state:
case a
src✝ : AddGroupWithOne ℂ := Complex.addGroupWithOne
a✝ b✝ c✝ : ℂ
⊢ a✝.re * (b✝.re + c✝.re) - a✝.im * (b✝.im + c✝.im) = a✝.re * b✝.re - a✝.im * b✝.im + (a✝.re * c✝.re - a✝.im * c✝.im)

Next tactic:
ring
instance commRing : CommRing ℂ :=
  { Complex.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec _ ⟨(1 : ℂ)⟩ ⟨(· * ·)⟩
    add_comm := by intros; ext <;> simp [add_comm]
    left_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> 

Proof state:
case a
src✝ : AddGroupWithOne ℂ := Complex.addGroupWithOne
a✝ b✝ c✝ : ℂ
⊢ a✝.re * (b✝.im + c✝.im) + a✝.im * (b✝.re + c✝.re) = a✝.re * b✝.im + a✝.im * b✝.re + (a✝.re * c✝.im + a✝.im * c✝.re)

Next tactic:
ring
instance commRing : CommRing ℂ :=
  { Complex.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec _ ⟨(1 : ℂ)⟩ ⟨(· * ·)⟩
    add_comm := by intros; ext <;> simp [add_comm]
    left_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> ring
    right_distrib := by
      

Proof state:
src✝ : AddGroupWithOne ℂ := Complex.addGroupWithOne
⊢ ∀ (a b c : ℂ), (a + b) * c = a * c + b * c

Next tactic:
intros
instance commRing : CommRing ℂ :=
  { Complex.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec _ ⟨(1 : ℂ)⟩ ⟨(· * ·)⟩
    add_comm := by intros; ext <;> simp [add_comm]
    left_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> ring
    right_distrib := by
      intros; 

Proof state:
src✝ : AddGroupWithOne ℂ := Complex.addGroupWithOne
a✝ b✝ c✝ : ℂ
⊢ (a✝ + b✝) * c✝ = a✝ * c✝ + b✝ * c✝

Next tactic:
ext
instance commRing : CommRing ℂ :=
  { Complex.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec _ ⟨(1 : ℂ)⟩ ⟨(· * ·)⟩
    add_comm := by intros; ext <;> simp [add_comm]
    left_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> ring
    right_distrib := by
      intros; ext <;> 

Proof state:
case a
src✝ : AddGroupWithOne ℂ := Complex.addGroupWithOne
a✝ b✝ c✝ : ℂ
⊢ ((a✝ + b✝) * c✝).re = (a✝ * c✝ + b✝ * c✝).re

Next tactic:
simp [mul_re, mul_im]
instance commRing : CommRing ℂ :=
  { Complex.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec _ ⟨(1 : ℂ)⟩ ⟨(· * ·)⟩
    add_comm := by intros; ext <;> simp [add_comm]
    left_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> ring
    right_distrib := by
      intros; ext <;> 

Proof state:
case a
src✝ : AddGroupWithOne ℂ := Complex.addGroupWithOne
a✝ b✝ c✝ : ℂ
⊢ ((a✝ + b✝) * c✝).im = (a✝ * c✝ + b✝ * c✝).im

Next tactic:
simp [mul_re, mul_im]
instance commRing : CommRing ℂ :=
  { Complex.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec _ ⟨(1 : ℂ)⟩ ⟨(· * ·)⟩
    add_comm := by intros; ext <;> simp [add_comm]
    left_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> ring
    right_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> 

Proof state:
case a
src✝ : AddGroupWithOne ℂ := Complex.addGroupWithOne
a✝ b✝ c✝ : ℂ
⊢ (a✝.re + b✝.re) * c✝.re - (a✝.im + b✝.im) * c✝.im = a✝.re * c✝.re - a✝.im * c✝.im + (b✝.re * c✝.re - b✝.im * c✝.im)

Next tactic:
ring
instance commRing : CommRing ℂ :=
  { Complex.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec _ ⟨(1 : ℂ)⟩ ⟨(· * ·)⟩
    add_comm := by intros; ext <;> simp [add_comm]
    left_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> ring
    right_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> 

Proof state:
case a
src✝ : AddGroupWithOne ℂ := Complex.addGroupWithOne
a✝ b✝ c✝ : ℂ
⊢ (a✝.re + b✝.re) * c✝.im + (a✝.im + b✝.im) * c✝.re = a✝.re * c✝.im + a✝.im * c✝.re + (b✝.re * c✝.im + b✝.im * c✝.re)

Next tactic:
ring
instance commRing : CommRing ℂ :=
  { Complex.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec _ ⟨(1 : ℂ)⟩ ⟨(· * ·)⟩
    add_comm := by intros; ext <;> simp [add_comm]
    left_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> ring
    right_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> ring
    zero_mul := by 

Proof state:
src✝ : AddGroupWithOne ℂ := Complex.addGroupWithOne
⊢ ∀ (a : ℂ), 0 * a = 0

Next tactic:
intros
instance commRing : CommRing ℂ :=
  { Complex.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec _ ⟨(1 : ℂ)⟩ ⟨(· * ·)⟩
    add_comm := by intros; ext <;> simp [add_comm]
    left_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> ring
    right_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> ring
    zero_mul := by intros; 

Proof state:
src✝ : AddGroupWithOne ℂ := Complex.addGroupWithOne
a✝ : ℂ
⊢ 0 * a✝ = 0

Next tactic:
ext
instance commRing : CommRing ℂ :=
  { Complex.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec _ ⟨(1 : ℂ)⟩ ⟨(· * ·)⟩
    add_comm := by intros; ext <;> simp [add_comm]
    left_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> ring
    right_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> ring
    zero_mul := by intros; ext <;> 

Proof state:
case a
src✝ : AddGroupWithOne ℂ := Complex.addGroupWithOne
a✝ : ℂ
⊢ (0 * a✝).re = 0.re

Next tactic:
simp [zero_mul]
instance commRing : CommRing ℂ :=
  { Complex.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec _ ⟨(1 : ℂ)⟩ ⟨(· * ·)⟩
    add_comm := by intros; ext <;> simp [add_comm]
    left_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> ring
    right_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> ring
    zero_mul := by intros; ext <;> 

Proof state:
case a
src✝ : AddGroupWithOne ℂ := Complex.addGroupWithOne
a✝ : ℂ
⊢ (0 * a✝).im = 0.im

Next tactic:
simp [zero_mul]
instance commRing : CommRing ℂ :=
  { Complex.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec _ ⟨(1 : ℂ)⟩ ⟨(· * ·)⟩
    add_comm := by intros; ext <;> simp [add_comm]
    left_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> ring
    right_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> ring
    zero_mul := by intros; ext <;> simp [zero_mul]
    mul_zero := by 

Proof state:
src✝ : AddGroupWithOne ℂ := Complex.addGroupWithOne
⊢ ∀ (a : ℂ), a * 0 = 0

Next tactic:
intros
instance commRing : CommRing ℂ :=
  { Complex.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec _ ⟨(1 : ℂ)⟩ ⟨(· * ·)⟩
    add_comm := by intros; ext <;> simp [add_comm]
    left_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> ring
    right_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> ring
    zero_mul := by intros; ext <;> simp [zero_mul]
    mul_zero := by intros; 

Proof state:
src✝ : AddGroupWithOne ℂ := Complex.addGroupWithOne
a✝ : ℂ
⊢ a✝ * 0 = 0

Next tactic:
ext
instance commRing : CommRing ℂ :=
  { Complex.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec _ ⟨(1 : ℂ)⟩ ⟨(· * ·)⟩
    add_comm := by intros; ext <;> simp [add_comm]
    left_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> ring
    right_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> ring
    zero_mul := by intros; ext <;> simp [zero_mul]
    mul_zero := by intros; ext <;> 

Proof state:
case a
src✝ : AddGroupWithOne ℂ := Complex.addGroupWithOne
a✝ : ℂ
⊢ (a✝ * 0).re = 0.re

Next tactic:
simp [mul_zero]
instance commRing : CommRing ℂ :=
  { Complex.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec _ ⟨(1 : ℂ)⟩ ⟨(· * ·)⟩
    add_comm := by intros; ext <;> simp [add_comm]
    left_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> ring
    right_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> ring
    zero_mul := by intros; ext <;> simp [zero_mul]
    mul_zero := by intros; ext <;> 

Proof state:
case a
src✝ : AddGroupWithOne ℂ := Complex.addGroupWithOne
a✝ : ℂ
⊢ (a✝ * 0).im = 0.im

Next tactic:
simp [mul_zero]
instance commRing : CommRing ℂ :=
  { Complex.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec _ ⟨(1 : ℂ)⟩ ⟨(· * ·)⟩
    add_comm := by intros; ext <;> simp [add_comm]
    left_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> ring
    right_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> ring
    zero_mul := by intros; ext <;> simp [zero_mul]
    mul_zero := by intros; ext <;> simp [mul_zero]
    mul_assoc := by 

Proof state:
src✝ : AddGroupWithOne ℂ := Complex.addGroupWithOne
⊢ ∀ (a b c : ℂ), a * b * c = a * (b * c)

Next tactic:
intros
instance commRing : CommRing ℂ :=
  { Complex.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec _ ⟨(1 : ℂ)⟩ ⟨(· * ·)⟩
    add_comm := by intros; ext <;> simp [add_comm]
    left_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> ring
    right_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> ring
    zero_mul := by intros; ext <;> simp [zero_mul]
    mul_zero := by intros; ext <;> simp [mul_zero]
    mul_assoc := by intros; 

Proof state:
src✝ : AddGroupWithOne ℂ := Complex.addGroupWithOne
a✝ b✝ c✝ : ℂ
⊢ a✝ * b✝ * c✝ = a✝ * (b✝ * c✝)

Next tactic:
ext
instance commRing : CommRing ℂ :=
  { Complex.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec _ ⟨(1 : ℂ)⟩ ⟨(· * ·)⟩
    add_comm := by intros; ext <;> simp [add_comm]
    left_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> ring
    right_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> ring
    zero_mul := by intros; ext <;> simp [zero_mul]
    mul_zero := by intros; ext <;> simp [mul_zero]
    mul_assoc := by intros; ext <;> 

Proof state:
case a
src✝ : AddGroupWithOne ℂ := Complex.addGroupWithOne
a✝ b✝ c✝ : ℂ
⊢ (a✝ * b✝ * c✝).re = (a✝ * (b✝ * c✝)).re

Next tactic:
simp [mul_assoc]
instance commRing : CommRing ℂ :=
  { Complex.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec _ ⟨(1 : ℂ)⟩ ⟨(· * ·)⟩
    add_comm := by intros; ext <;> simp [add_comm]
    left_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> ring
    right_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> ring
    zero_mul := by intros; ext <;> simp [zero_mul]
    mul_zero := by intros; ext <;> simp [mul_zero]
    mul_assoc := by intros; ext <;> 

Proof state:
case a
src✝ : AddGroupWithOne ℂ := Complex.addGroupWithOne
a✝ b✝ c✝ : ℂ
⊢ (a✝ * b✝ * c✝).im = (a✝ * (b✝ * c✝)).im

Next tactic:
simp [mul_assoc]
instance commRing : CommRing ℂ :=
  { Complex.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec _ ⟨(1 : ℂ)⟩ ⟨(· * ·)⟩
    add_comm := by intros; ext <;> simp [add_comm]
    left_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> ring
    right_distrib := by
      intros; ext <;> simp [mul_re, mul_im] <;> ring
    zero_mul := by intros; ext <;> simp [zero_mul]
    mul_zero := by intros; ext <;> simp [mul_zero]
    mul_assoc := by intros; ext <;> simp [mul_assoc] <;> 

Proof state:
case a
src✝ : AddGroupWithOne ℂ := Complex.addGroupWithOne
a✝ b✝ c✝ : ℂ
⊢ (a✝.re * b✝.re - a✝.im * b✝.im) * c✝.re - (a✝.re * b✝.im + a✝.im * b✝.re) * c✝.im =
    a✝.re * (b✝.re * c✝.re - b✝.im * c✝.im) - a✝.im * (b✝.re * c✝.im + b✝.im * c✝.re)

Next tactic:
ring