l3lab/ntp-mathlib huggingface dataset:
theorem ext : ∀ {z w : ℤ√d}, z = w ↔ z.re = w.re ∧ z.im = w.im
  | ⟨x, y⟩, ⟨x', y'⟩ =>
    ⟨fun h => by injection h; constructor <;> assumption,
     fun ⟨h₁, h₂⟩ => by 

Proof state:
d x y x' y' : ℤ
x✝ : { re := x, im := y }.re = { re := x', im := y' }.re ∧ { re := x, im := y }.im = { re := x', im := y' }.im
h₁ : { re := x, im := y }.re = { re := x', im := y' }.re
h₂ : { re := x, im := y }.im = { re := x', im := y' }.im
⊢ { re := x, im := y } = { re := x', im := y' }

Next tactic:
congr
@[simp]
theorem bit1_im (z) : (bit1 z : ℤ√d).im = bit0 z.im := by 

Proof state:
d : ℤ
z : ℤ√d
⊢ (bit1 z).im = bit0 z.im

Next tactic:
simp [bit1]
instance addCommGroup : AddCommGroup (ℤ√d) := by
  

Proof state:
d : ℤ
⊢ AddCommGroup (ℤ√d)

Next tactic:
refine
  { add := (· + ·)
    zero := (0 : ℤ√d)
    sub := fun a b => a + -b
    neg := Neg.neg
    zsmul := @zsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩ ⟨Neg.neg⟩
    nsmul := @nsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩
    add_assoc := ?_
    zero_add := ?_
    add_zero := ?_
    add_left_neg := ?_
    add_comm := ?_ }
instance addCommGroup : AddCommGroup (ℤ√d) := by
  refine
  { add := (· + ·)
    zero := (0 : ℤ√d)
    sub := fun a b => a + -b
    neg := Neg.neg
    zsmul := @zsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩ ⟨Neg.neg⟩
    nsmul := @nsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩
    add_assoc := ?_
    zero_add := ?_
    add_zero := ?_
    add_left_neg := ?_
    add_comm := ?_ } <;>
  

Proof state:
case refine_1
d : ℤ
⊢ ∀ (a b c : ℤ√d), a + b + c = a + (b + c)

Next tactic:
intros
instance addCommGroup : AddCommGroup (ℤ√d) := by
  refine
  { add := (· + ·)
    zero := (0 : ℤ√d)
    sub := fun a b => a + -b
    neg := Neg.neg
    zsmul := @zsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩ ⟨Neg.neg⟩
    nsmul := @nsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩
    add_assoc := ?_
    zero_add := ?_
    add_zero := ?_
    add_left_neg := ?_
    add_comm := ?_ } <;>
  

Proof state:
case refine_2
d : ℤ
⊢ ∀ (a : ℤ√d), 0 + a = a

Next tactic:
intros
instance addCommGroup : AddCommGroup (ℤ√d) := by
  refine
  { add := (· + ·)
    zero := (0 : ℤ√d)
    sub := fun a b => a + -b
    neg := Neg.neg
    zsmul := @zsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩ ⟨Neg.neg⟩
    nsmul := @nsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩
    add_assoc := ?_
    zero_add := ?_
    add_zero := ?_
    add_left_neg := ?_
    add_comm := ?_ } <;>
  

Proof state:
case refine_3
d : ℤ
⊢ ∀ (a : ℤ√d), a + 0 = a

Next tactic:
intros
instance addCommGroup : AddCommGroup (ℤ√d) := by
  refine
  { add := (· + ·)
    zero := (0 : ℤ√d)
    sub := fun a b => a + -b
    neg := Neg.neg
    zsmul := @zsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩ ⟨Neg.neg⟩
    nsmul := @nsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩
    add_assoc := ?_
    zero_add := ?_
    add_zero := ?_
    add_left_neg := ?_
    add_comm := ?_ } <;>
  

Proof state:
case refine_4
d : ℤ
⊢ ∀ (a : ℤ√d), -a + a = 0

Next tactic:
intros
instance addCommGroup : AddCommGroup (ℤ√d) := by
  refine
  { add := (· + ·)
    zero := (0 : ℤ√d)
    sub := fun a b => a + -b
    neg := Neg.neg
    zsmul := @zsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩ ⟨Neg.neg⟩
    nsmul := @nsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩
    add_assoc := ?_
    zero_add := ?_
    add_zero := ?_
    add_left_neg := ?_
    add_comm := ?_ } <;>
  

Proof state:
case refine_5
d : ℤ
⊢ ∀ (a b : ℤ√d), a + b = b + a

Next tactic:
intros
instance addCommGroup : AddCommGroup (ℤ√d) := by
  refine
  { add := (· + ·)
    zero := (0 : ℤ√d)
    sub := fun a b => a + -b
    neg := Neg.neg
    zsmul := @zsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩ ⟨Neg.neg⟩
    nsmul := @nsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩
    add_assoc := ?_
    zero_add := ?_
    add_zero := ?_
    add_left_neg := ?_
    add_comm := ?_ } <;>
  intros <;>
  

Proof state:
case refine_1
d : ℤ
a✝ b✝ c✝ : ℤ√d
⊢ a✝ + b✝ + c✝ = a✝ + (b✝ + c✝)

Next tactic:
simp [ext, add_comm, add_left_comm]
instance addCommGroup : AddCommGroup (ℤ√d) := by
  refine
  { add := (· + ·)
    zero := (0 : ℤ√d)
    sub := fun a b => a + -b
    neg := Neg.neg
    zsmul := @zsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩ ⟨Neg.neg⟩
    nsmul := @nsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩
    add_assoc := ?_
    zero_add := ?_
    add_zero := ?_
    add_left_neg := ?_
    add_comm := ?_ } <;>
  intros <;>
  

Proof state:
case refine_2
d : ℤ
a✝ : ℤ√d
⊢ 0 + a✝ = a✝

Next tactic:
simp [ext, add_comm, add_left_comm]
instance addCommGroup : AddCommGroup (ℤ√d) := by
  refine
  { add := (· + ·)
    zero := (0 : ℤ√d)
    sub := fun a b => a + -b
    neg := Neg.neg
    zsmul := @zsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩ ⟨Neg.neg⟩
    nsmul := @nsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩
    add_assoc := ?_
    zero_add := ?_
    add_zero := ?_
    add_left_neg := ?_
    add_comm := ?_ } <;>
  intros <;>
  

Proof state:
case refine_3
d : ℤ
a✝ : ℤ√d
⊢ a✝ + 0 = a✝

Next tactic:
simp [ext, add_comm, add_left_comm]
instance addCommGroup : AddCommGroup (ℤ√d) := by
  refine
  { add := (· + ·)
    zero := (0 : ℤ√d)
    sub := fun a b => a + -b
    neg := Neg.neg
    zsmul := @zsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩ ⟨Neg.neg⟩
    nsmul := @nsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩
    add_assoc := ?_
    zero_add := ?_
    add_zero := ?_
    add_left_neg := ?_
    add_comm := ?_ } <;>
  intros <;>
  

Proof state:
case refine_4
d : ℤ
a✝ : ℤ√d
⊢ -a✝ + a✝ = 0

Next tactic:
simp [ext, add_comm, add_left_comm]
instance addCommGroup : AddCommGroup (ℤ√d) := by
  refine
  { add := (· + ·)
    zero := (0 : ℤ√d)
    sub := fun a b => a + -b
    neg := Neg.neg
    zsmul := @zsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩ ⟨Neg.neg⟩
    nsmul := @nsmulRec (ℤ√d) ⟨0⟩ ⟨(· + ·)⟩
    add_assoc := ?_
    zero_add := ?_
    add_zero := ?_
    add_left_neg := ?_
    add_comm := ?_ } <;>
  intros <;>
  

Proof state:
case refine_5
d : ℤ
a✝ b✝ : ℤ√d
⊢ a✝ + b✝ = b✝ + a✝

Next tactic:
simp [ext, add_comm, add_left_comm]
instance commRing : CommRing (ℤ√d) := by
  

Proof state:
d : ℤ
⊢ CommRing (ℤ√d)

Next tactic:
refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ }
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  

Proof state:
case refine_1
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
⊢ ∀ (a b : ℤ√d), a + b = b + a

Next tactic:
intros
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  

Proof state:
case refine_2
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
⊢ ∀ (a b c : ℤ√d), a * (b + c) = a * b + a * c

Next tactic:
intros
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  

Proof state:
case refine_3
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
⊢ ∀ (a b c : ℤ√d), (a + b) * c = a * c + b * c

Next tactic:
intros
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  

Proof state:
case refine_4
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
⊢ ∀ (a : ℤ√d), 0 * a = 0

Next tactic:
intros
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  

Proof state:
case refine_5
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
⊢ ∀ (a : ℤ√d), a * 0 = 0

Next tactic:
intros
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  

Proof state:
case refine_6
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
⊢ ∀ (a b c : ℤ√d), a * b * c = a * (b * c)

Next tactic:
intros
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  

Proof state:
case refine_7
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
⊢ ∀ (a : ℤ√d), 1 * a = a

Next tactic:
intros
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  

Proof state:
case refine_8
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
⊢ ∀ (a : ℤ√d), a * 1 = a

Next tactic:
intros
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  

Proof state:
case refine_9
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
⊢ ∀ (a b : ℤ√d), a * b = b * a

Next tactic:
intros
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  intros <;>
  

Proof state:
case refine_1
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
a✝ b✝ : ℤ√d
⊢ a✝ + b✝ = b✝ + a✝

Next tactic:
refine ext.mpr ⟨?_, ?_⟩
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  intros <;>
  

Proof state:
case refine_2
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
a✝ b✝ c✝ : ℤ√d
⊢ a✝ * (b✝ + c✝) = a✝ * b✝ + a✝ * c✝

Next tactic:
refine ext.mpr ⟨?_, ?_⟩
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  intros <;>
  

Proof state:
case refine_3
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
a✝ b✝ c✝ : ℤ√d
⊢ (a✝ + b✝) * c✝ = a✝ * c✝ + b✝ * c✝

Next tactic:
refine ext.mpr ⟨?_, ?_⟩
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  intros <;>
  

Proof state:
case refine_4
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
a✝ : ℤ√d
⊢ 0 * a✝ = 0

Next tactic:
refine ext.mpr ⟨?_, ?_⟩
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  intros <;>
  

Proof state:
case refine_5
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
a✝ : ℤ√d
⊢ a✝ * 0 = 0

Next tactic:
refine ext.mpr ⟨?_, ?_⟩
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  intros <;>
  

Proof state:
case refine_6
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
a✝ b✝ c✝ : ℤ√d
⊢ a✝ * b✝ * c✝ = a✝ * (b✝ * c✝)

Next tactic:
refine ext.mpr ⟨?_, ?_⟩
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  intros <;>
  

Proof state:
case refine_7
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
a✝ : ℤ√d
⊢ 1 * a✝ = a✝

Next tactic:
refine ext.mpr ⟨?_, ?_⟩
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  intros <;>
  

Proof state:
case refine_8
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
a✝ : ℤ√d
⊢ a✝ * 1 = a✝

Next tactic:
refine ext.mpr ⟨?_, ?_⟩
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  intros <;>
  

Proof state:
case refine_9
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
a✝ b✝ : ℤ√d
⊢ a✝ * b✝ = b✝ * a✝

Next tactic:
refine ext.mpr ⟨?_, ?_⟩
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  intros <;>
  refine ext.mpr ⟨?_, ?_⟩ <;>
  

Proof state:
case refine_1.refine_1
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
a✝ b✝ : ℤ√d
⊢ (a✝ + b✝).re = (b✝ + a✝).re

Next tactic:
simp
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  intros <;>
  refine ext.mpr ⟨?_, ?_⟩ <;>
  

Proof state:
case refine_1.refine_2
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
a✝ b✝ : ℤ√d
⊢ (a✝ + b✝).im = (b✝ + a✝).im

Next tactic:
simp
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  intros <;>
  refine ext.mpr ⟨?_, ?_⟩ <;>
  

Proof state:
case refine_2.refine_1
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
a✝ b✝ c✝ : ℤ√d
⊢ (a✝ * (b✝ + c✝)).re = (a✝ * b✝ + a✝ * c✝).re

Next tactic:
simp
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  intros <;>
  refine ext.mpr ⟨?_, ?_⟩ <;>
  

Proof state:
case refine_2.refine_2
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
a✝ b✝ c✝ : ℤ√d
⊢ (a✝ * (b✝ + c✝)).im = (a✝ * b✝ + a✝ * c✝).im

Next tactic:
simp
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  intros <;>
  refine ext.mpr ⟨?_, ?_⟩ <;>
  

Proof state:
case refine_3.refine_1
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
a✝ b✝ c✝ : ℤ√d
⊢ ((a✝ + b✝) * c✝).re = (a✝ * c✝ + b✝ * c✝).re

Next tactic:
simp
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  intros <;>
  refine ext.mpr ⟨?_, ?_⟩ <;>
  

Proof state:
case refine_3.refine_2
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
a✝ b✝ c✝ : ℤ√d
⊢ ((a✝ + b✝) * c✝).im = (a✝ * c✝ + b✝ * c✝).im

Next tactic:
simp
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  intros <;>
  refine ext.mpr ⟨?_, ?_⟩ <;>
  

Proof state:
case refine_4.refine_1
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
a✝ : ℤ√d
⊢ (0 * a✝).re = 0.re

Next tactic:
simp
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  intros <;>
  refine ext.mpr ⟨?_, ?_⟩ <;>
  

Proof state:
case refine_4.refine_2
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
a✝ : ℤ√d
⊢ (0 * a✝).im = 0.im

Next tactic:
simp
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  intros <;>
  refine ext.mpr ⟨?_, ?_⟩ <;>
  

Proof state:
case refine_5.refine_1
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
a✝ : ℤ√d
⊢ (a✝ * 0).re = 0.re

Next tactic:
simp
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  intros <;>
  refine ext.mpr ⟨?_, ?_⟩ <;>
  

Proof state:
case refine_5.refine_2
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
a✝ : ℤ√d
⊢ (a✝ * 0).im = 0.im

Next tactic:
simp
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  intros <;>
  refine ext.mpr ⟨?_, ?_⟩ <;>
  

Proof state:
case refine_6.refine_1
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
a✝ b✝ c✝ : ℤ√d
⊢ (a✝ * b✝ * c✝).re = (a✝ * (b✝ * c✝)).re

Next tactic:
simp
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  intros <;>
  refine ext.mpr ⟨?_, ?_⟩ <;>
  

Proof state:
case refine_6.refine_2
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
a✝ b✝ c✝ : ℤ√d
⊢ (a✝ * b✝ * c✝).im = (a✝ * (b✝ * c✝)).im

Next tactic:
simp
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  intros <;>
  refine ext.mpr ⟨?_, ?_⟩ <;>
  

Proof state:
case refine_7.refine_1
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
a✝ : ℤ√d
⊢ (1 * a✝).re = a✝.re

Next tactic:
simp
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  intros <;>
  refine ext.mpr ⟨?_, ?_⟩ <;>
  

Proof state:
case refine_7.refine_2
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
a✝ : ℤ√d
⊢ (1 * a✝).im = a✝.im

Next tactic:
simp
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  intros <;>
  refine ext.mpr ⟨?_, ?_⟩ <;>
  

Proof state:
case refine_8.refine_1
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
a✝ : ℤ√d
⊢ (a✝ * 1).re = a✝.re

Next tactic:
simp
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  intros <;>
  refine ext.mpr ⟨?_, ?_⟩ <;>
  

Proof state:
case refine_8.refine_2
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
a✝ : ℤ√d
⊢ (a✝ * 1).im = a✝.im

Next tactic:
simp
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  intros <;>
  refine ext.mpr ⟨?_, ?_⟩ <;>
  

Proof state:
case refine_9.refine_1
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
a✝ b✝ : ℤ√d
⊢ (a✝ * b✝).re = (b✝ * a✝).re

Next tactic:
simp
instance commRing : CommRing (ℤ√d) := by
  refine
  { Zsqrtd.addGroupWithOne with
    mul := (· * ·)
    npow := @npowRec (ℤ√d) ⟨1⟩ ⟨(· * ·)⟩,
    add_comm := ?_
    left_distrib := ?_
    right_distrib := ?_
    zero_mul := ?_
    mul_zero := ?_
    mul_assoc := ?_
    one_mul := ?_
    mul_one := ?_
    mul_comm := ?_ } <;>
  intros <;>
  refine ext.mpr ⟨?_, ?_⟩ <;>
  

Proof state:
case refine_9.refine_2
d : ℤ
src✝ : AddGroupWithOne (ℤ√d) := addGroupWithOne
a✝ b✝ : ℤ√d
⊢ (a✝ * b✝).im = (b✝ * a✝).im

Next tactic:
simp