l3lab/ntp-mathlib huggingface dataset:
theorem abs_re_le_norm (z : K) : |re z| ≤ ‖z‖ := by
  rw [mul_self_le_mul_self_iff (abs_nonneg _) (norm_nonneg _), abs_mul_abs_self, mul_self_norm]
  

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
z : K
⊢ re z * re z ≤ normSq z

Next tactic:
apply re_sq_le_normSq
theorem abs_im_le_norm (z : K) : |im z| ≤ ‖z‖ := by
  

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
z : K
⊢ |im z| ≤ ‖z‖

Next tactic:
rw [mul_self_le_mul_self_iff (abs_nonneg _) (norm_nonneg _), abs_mul_abs_self, mul_self_norm]
theorem abs_im_le_norm (z : K) : |im z| ≤ ‖z‖ := by
  rw [mul_self_le_mul_self_iff (abs_nonneg _) (norm_nonneg _), abs_mul_abs_self, mul_self_norm]
  

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
z : K
⊢ im z * im z ≤ normSq z

Next tactic:
apply im_sq_le_normSq
theorem im_eq_zero_of_le {a : K} (h : ‖a‖ ≤ re a) : im a = 0 := by
  

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
a : K
h : ‖a‖ ≤ re a
⊢ im a = 0

Next tactic:
simpa only [mul_self_norm a, normSq_apply, self_eq_add_right, mul_self_eq_zero]
    using congr_arg (fun z => z * z) ((re_le_norm a).antisymm h)
theorem re_eq_self_of_le {a : K} (h : ‖a‖ ≤ re a) : (re a : K) = a := by
  

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
a : K
h : ‖a‖ ≤ re a
⊢ ↑(re a) = a

Next tactic:
rw [← conj_eq_iff_re, conj_eq_iff_im, im_eq_zero_of_le h]
theorem abs_re_div_norm_le_one (z : K) : |re z / ‖z‖| ≤ 1 := by
  

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
z : K
⊢ |re z / ‖z‖| ≤ 1

Next tactic:
rw [abs_div, abs_norm]
theorem abs_re_div_norm_le_one (z : K) : |re z / ‖z‖| ≤ 1 := by
  rw [abs_div, abs_norm]
  

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
z : K
⊢ |re z| / ‖z‖ ≤ 1

Next tactic:
exact div_le_one_of_le (abs_re_le_norm _) (norm_nonneg _)
theorem abs_im_div_norm_le_one (z : K) : |im z / ‖z‖| ≤ 1 := by
  

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
z : K
⊢ |im z / ‖z‖| ≤ 1

Next tactic:
rw [abs_div, abs_norm]
theorem abs_im_div_norm_le_one (z : K) : |im z / ‖z‖| ≤ 1 := by
  rw [abs_div, abs_norm]
  

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
z : K
⊢ |im z| / ‖z‖ ≤ 1

Next tactic:
exact div_le_one_of_le (abs_im_le_norm _) (norm_nonneg _)
theorem norm_I_of_ne_zero (hI : (I : K) ≠ 0) : ‖(I : K)‖ = 1 := by
  

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
hI : I ≠ 0
⊢ ‖I‖ = 1

Next tactic:
rw [← mul_self_inj_of_nonneg (norm_nonneg I) zero_le_one, one_mul, ← norm_mul,
    I_mul_I_of_nonzero hI, norm_neg, norm_one]
theorem re_eq_norm_of_mul_conj (x : K) : re (x * conj x) = ‖x * conj x‖ := by
  

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
x : K
⊢ re (x * (starRingEnd K) x) = ‖x * (starRingEnd K) x‖

Next tactic:
rw [mul_conj, ofReal_re, norm_ofReal, abs_of_nonneg (normSq_nonneg _)]
theorem norm_sq_re_add_conj (x : K) : ‖x + conj x‖ ^ 2 = re (x + conj x) ^ 2 := by
  

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
x : K
⊢ ‖x + (starRingEnd K) x‖ ^ 2 = re (x + (starRingEnd K) x) ^ 2

Next tactic:
rw [add_conj, ← ofReal_ofNat, ← ofReal_mul, norm_ofReal, sq_abs, ofReal_re]
theorem norm_sq_re_conj_add (x : K) : ‖conj x + x‖ ^ 2 = re (conj x + x) ^ 2 := by
  

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
x : K
⊢ ‖(starRingEnd K) x + x‖ ^ 2 = re ((starRingEnd K) x + x) ^ 2

Next tactic:
rw [add_comm, norm_sq_re_add_conj]
theorem isCauSeq_re (f : CauSeq K norm) : IsCauSeq abs fun n => re (f n) := fun ε ε0 =>
  (f.cauchy ε0).imp fun i H j ij =>
    lt_of_le_of_lt (by 

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
f : CauSeq K norm
ε : ℝ
ε0 : ε > 0
i : ℕ
H : ∀ j ≥ i, ‖↑f j - ↑f i‖ < ε
j : ℕ
ij : j ≥ i
⊢ |(fun n => re (↑f n)) j - (fun n => re (↑f n)) i| ≤ ‖↑f j - ↑f i‖

Next tactic:
simpa only [map_sub] using abs_re_le_norm (f j - f i)
theorem isCauSeq_im (f : CauSeq K norm) : IsCauSeq abs fun n => im (f n) := fun ε ε0 =>
  (f.cauchy ε0).imp fun i H j ij =>
    lt_of_le_of_lt (by 

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
f : CauSeq K norm
ε : ℝ
ε0 : ε > 0
i : ℕ
H : ∀ j ≥ i, ‖↑f j - ↑f i‖ < ε
j : ℕ
ij : j ≥ i
⊢ |(fun n => im (↑f n)) j - (fun n => im (↑f n)) i| ≤ ‖↑f j - ↑f i‖

Next tactic:
simpa only [map_sub] using abs_im_le_norm (f j - f i)
noncomputable instance Real.isROrC : IsROrC ℝ where
  re := AddMonoidHom.id ℝ
  im := 0
  I := 0
  I_re_ax := by 

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
⊢ (AddMonoidHom.id ℝ) 0 = 0

Next tactic:
simp only [AddMonoidHom.map_zero]
noncomputable instance Real.isROrC : IsROrC ℝ where
  re := AddMonoidHom.id ℝ
  im := 0
  I := 0
  I_re_ax := by simp only [AddMonoidHom.map_zero]
  I_mul_I_ax := Or.intro_left _ rfl
  re_add_im_ax z := by
    

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
z : ℝ
⊢ (algebraMap ℝ ℝ) ((AddMonoidHom.id ℝ) z) + (algebraMap ℝ ℝ) (0 z) * 0 = z

Next tactic:
simp only [add_zero, mul_zero, Algebra.id.map_eq_id, RingHom.id_apply, AddMonoidHom.id_apply]
noncomputable instance Real.isROrC : IsROrC ℝ where
  re := AddMonoidHom.id ℝ
  im := 0
  I := 0
  I_re_ax := by simp only [AddMonoidHom.map_zero]
  I_mul_I_ax := Or.intro_left _ rfl
  re_add_im_ax z := by
    simp only [add_zero, mul_zero, Algebra.id.map_eq_id, RingHom.id_apply, AddMonoidHom.id_apply]
  ofReal_re_ax f := rfl
  ofReal_im_ax r := rfl
  mul_re_ax z w := by 

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
z w : ℝ
⊢ (AddMonoidHom.id ℝ) (z * w) = (AddMonoidHom.id ℝ) z * (AddMonoidHom.id ℝ) w - 0 z * 0 w

Next tactic:
simp only [sub_zero, mul_zero, AddMonoidHom.zero_apply, AddMonoidHom.id_apply]
noncomputable instance Real.isROrC : IsROrC ℝ where
  re := AddMonoidHom.id ℝ
  im := 0
  I := 0
  I_re_ax := by simp only [AddMonoidHom.map_zero]
  I_mul_I_ax := Or.intro_left _ rfl
  re_add_im_ax z := by
    simp only [add_zero, mul_zero, Algebra.id.map_eq_id, RingHom.id_apply, AddMonoidHom.id_apply]
  ofReal_re_ax f := rfl
  ofReal_im_ax r := rfl
  mul_re_ax z w := by simp only [sub_zero, mul_zero, AddMonoidHom.zero_apply, AddMonoidHom.id_apply]
  mul_im_ax z w := by 

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
z w : ℝ
⊢ 0 (z * w) = (AddMonoidHom.id ℝ) z * 0 w + 0 z * (AddMonoidHom.id ℝ) w

Next tactic:
simp only [add_zero, zero_mul, mul_zero, AddMonoidHom.zero_apply]
noncomputable instance Real.isROrC : IsROrC ℝ where
  re := AddMonoidHom.id ℝ
  im := 0
  I := 0
  I_re_ax := by simp only [AddMonoidHom.map_zero]
  I_mul_I_ax := Or.intro_left _ rfl
  re_add_im_ax z := by
    simp only [add_zero, mul_zero, Algebra.id.map_eq_id, RingHom.id_apply, AddMonoidHom.id_apply]
  ofReal_re_ax f := rfl
  ofReal_im_ax r := rfl
  mul_re_ax z w := by simp only [sub_zero, mul_zero, AddMonoidHom.zero_apply, AddMonoidHom.id_apply]
  mul_im_ax z w := by simp only [add_zero, zero_mul, mul_zero, AddMonoidHom.zero_apply]
  conj_re_ax z := by 

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
z : ℝ
⊢ (AddMonoidHom.id ℝ) ((starRingEnd ℝ) z) = (AddMonoidHom.id ℝ) z

Next tactic:
simp only [starRingEnd_apply, star_id_of_comm]
noncomputable instance Real.isROrC : IsROrC ℝ where
  re := AddMonoidHom.id ℝ
  im := 0
  I := 0
  I_re_ax := by simp only [AddMonoidHom.map_zero]
  I_mul_I_ax := Or.intro_left _ rfl
  re_add_im_ax z := by
    simp only [add_zero, mul_zero, Algebra.id.map_eq_id, RingHom.id_apply, AddMonoidHom.id_apply]
  ofReal_re_ax f := rfl
  ofReal_im_ax r := rfl
  mul_re_ax z w := by simp only [sub_zero, mul_zero, AddMonoidHom.zero_apply, AddMonoidHom.id_apply]
  mul_im_ax z w := by simp only [add_zero, zero_mul, mul_zero, AddMonoidHom.zero_apply]
  conj_re_ax z := by simp only [starRingEnd_apply, star_id_of_comm]
  conj_im_ax _ := by 

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
x✝ : ℝ
⊢ 0 ((starRingEnd ℝ) x✝) = -0 x✝

Next tactic:
simp only [neg_zero, AddMonoidHom.zero_apply]
noncomputable instance Real.isROrC : IsROrC ℝ where
  re := AddMonoidHom.id ℝ
  im := 0
  I := 0
  I_re_ax := by simp only [AddMonoidHom.map_zero]
  I_mul_I_ax := Or.intro_left _ rfl
  re_add_im_ax z := by
    simp only [add_zero, mul_zero, Algebra.id.map_eq_id, RingHom.id_apply, AddMonoidHom.id_apply]
  ofReal_re_ax f := rfl
  ofReal_im_ax r := rfl
  mul_re_ax z w := by simp only [sub_zero, mul_zero, AddMonoidHom.zero_apply, AddMonoidHom.id_apply]
  mul_im_ax z w := by simp only [add_zero, zero_mul, mul_zero, AddMonoidHom.zero_apply]
  conj_re_ax z := by simp only [starRingEnd_apply, star_id_of_comm]
  conj_im_ax _ := by simp only [neg_zero, AddMonoidHom.zero_apply]
  conj_I_ax := by 

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
⊢ (starRingEnd ℝ) 0 = -0

Next tactic:
simp only [RingHom.map_zero, neg_zero]
noncomputable instance Real.isROrC : IsROrC ℝ where
  re := AddMonoidHom.id ℝ
  im := 0
  I := 0
  I_re_ax := by simp only [AddMonoidHom.map_zero]
  I_mul_I_ax := Or.intro_left _ rfl
  re_add_im_ax z := by
    simp only [add_zero, mul_zero, Algebra.id.map_eq_id, RingHom.id_apply, AddMonoidHom.id_apply]
  ofReal_re_ax f := rfl
  ofReal_im_ax r := rfl
  mul_re_ax z w := by simp only [sub_zero, mul_zero, AddMonoidHom.zero_apply, AddMonoidHom.id_apply]
  mul_im_ax z w := by simp only [add_zero, zero_mul, mul_zero, AddMonoidHom.zero_apply]
  conj_re_ax z := by simp only [starRingEnd_apply, star_id_of_comm]
  conj_im_ax _ := by simp only [neg_zero, AddMonoidHom.zero_apply]
  conj_I_ax := by simp only [RingHom.map_zero, neg_zero]
  norm_sq_eq_def_ax z := by 

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
z : ℝ
⊢ ‖z‖ ^ 2 = (AddMonoidHom.id ℝ) z * (AddMonoidHom.id ℝ) z + 0 z * 0 z

Next tactic:
simp only [sq, Real.norm_eq_abs, ← abs_mul, abs_mul_self z, add_zero,
    mul_zero, AddMonoidHom.zero_apply, AddMonoidHom.id_apply]
noncomputable instance Real.isROrC : IsROrC ℝ where
  re := AddMonoidHom.id ℝ
  im := 0
  I := 0
  I_re_ax := by simp only [AddMonoidHom.map_zero]
  I_mul_I_ax := Or.intro_left _ rfl
  re_add_im_ax z := by
    simp only [add_zero, mul_zero, Algebra.id.map_eq_id, RingHom.id_apply, AddMonoidHom.id_apply]
  ofReal_re_ax f := rfl
  ofReal_im_ax r := rfl
  mul_re_ax z w := by simp only [sub_zero, mul_zero, AddMonoidHom.zero_apply, AddMonoidHom.id_apply]
  mul_im_ax z w := by simp only [add_zero, zero_mul, mul_zero, AddMonoidHom.zero_apply]
  conj_re_ax z := by simp only [starRingEnd_apply, star_id_of_comm]
  conj_im_ax _ := by simp only [neg_zero, AddMonoidHom.zero_apply]
  conj_I_ax := by simp only [RingHom.map_zero, neg_zero]
  norm_sq_eq_def_ax z := by simp only [sq, Real.norm_eq_abs, ← abs_mul, abs_mul_self z, add_zero,
    mul_zero, AddMonoidHom.zero_apply, AddMonoidHom.id_apply]
  mul_im_I_ax _ := by 

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
x✝ : ℝ
⊢ 0 x✝ * 0 0 = 0 x✝

Next tactic:
simp only [mul_zero, AddMonoidHom.zero_apply]
theorem lt_iff_re_im {z w : K} : z < w ↔ re z < re w ∧ im z = im w := by
  

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
z w : K
⊢ z < w ↔ re z < re w ∧ im z = im w

Next tactic:
simp_rw [lt_iff_le_and_ne, @IsROrC.le_iff_re_im K]
theorem lt_iff_re_im {z w : K} : z < w ↔ re z < re w ∧ im z = im w := by
  simp_rw [lt_iff_le_and_ne, @IsROrC.le_iff_re_im K]
  

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
z w : K
⊢ (re z ≤ re w ∧ im z = im w) ∧ z ≠ w ↔ (re z ≤ re w ∧ re z ≠ re w) ∧ im z = im w

Next tactic:
constructor
theorem lt_iff_re_im {z w : K} : z < w ↔ re z < re w ∧ im z = im w := by
  simp_rw [lt_iff_le_and_ne, @IsROrC.le_iff_re_im K]
  constructor
  · 

Proof state:
case mp
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
z w : K
⊢ (re z ≤ re w ∧ im z = im w) ∧ z ≠ w → (re z ≤ re w ∧ re z ≠ re w) ∧ im z = im w

Next tactic:
rintro ⟨⟨hr, hi⟩, heq⟩
theorem lt_iff_re_im {z w : K} : z < w ↔ re z < re w ∧ im z = im w := by
  simp_rw [lt_iff_le_and_ne, @IsROrC.le_iff_re_im K]
  constructor
  · rintro ⟨⟨hr, hi⟩, heq⟩
    

Proof state:
case mp.intro.intro
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
z w : K
heq : z ≠ w
hr : re z ≤ re w
hi : im z = im w
⊢ (re z ≤ re w ∧ re z ≠ re w) ∧ im z = im w

Next tactic:
exact ⟨⟨hr, mt (fun hreq => ext hreq hi) heq⟩, hi⟩
theorem lt_iff_re_im {z w : K} : z < w ↔ re z < re w ∧ im z = im w := by
  simp_rw [lt_iff_le_and_ne, @IsROrC.le_iff_re_im K]
  constructor
  · rintro ⟨⟨hr, hi⟩, heq⟩
    exact ⟨⟨hr, mt (fun hreq => ext hreq hi) heq⟩, hi⟩
  · 

Proof state:
case mpr
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
z w : K
⊢ (re z ≤ re w ∧ re z ≠ re w) ∧ im z = im w → (re z ≤ re w ∧ im z = im w) ∧ z ≠ w

Next tactic:
rintro ⟨⟨hr, hrn⟩, hi⟩
theorem lt_iff_re_im {z w : K} : z < w ↔ re z < re w ∧ im z = im w := by
  simp_rw [lt_iff_le_and_ne, @IsROrC.le_iff_re_im K]
  constructor
  · rintro ⟨⟨hr, hi⟩, heq⟩
    exact ⟨⟨hr, mt (fun hreq => ext hreq hi) heq⟩, hi⟩
  · rintro ⟨⟨hr, hrn⟩, hi⟩
    

Proof state:
case mpr.intro.intro
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
z w : K
hi : im z = im w
hr : re z ≤ re w
hrn : re z ≠ re w
⊢ (re z ≤ re w ∧ im z = im w) ∧ z ≠ w

Next tactic:
exact ⟨⟨hr, hi⟩, ne_of_apply_ne _ hrn⟩
theorem nonneg_iff {z : K} : 0 ≤ z ↔ 0 ≤ re z ∧ im z = 0 := by
  

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
z : K
⊢ 0 ≤ z ↔ 0 ≤ re z ∧ im z = 0

Next tactic:
simpa only [map_zero, eq_comm] using le_iff_re_im (z := 0) (w := z)
theorem pos_iff {z : K} : 0 < z ↔ 0 < re z ∧ im z = 0 := by
  

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
z : K
⊢ 0 < z ↔ 0 < re z ∧ im z = 0

Next tactic:
simpa only [map_zero, eq_comm] using lt_iff_re_im (z := 0) (w := z)
theorem nonpos_iff {z : K} : z ≤ 0 ↔ re z ≤ 0 ∧ im z = 0 := by
  

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
z : K
⊢ z ≤ 0 ↔ re z ≤ 0 ∧ im z = 0

Next tactic:
simpa only [map_zero] using le_iff_re_im (z := z) (w := 0)
theorem neg_iff {z : K} : z < 0 ↔ re z < 0 ∧ im z = 0 := by
  

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
z : K
⊢ z < 0 ↔ re z < 0 ∧ im z = 0

Next tactic:
simpa only [map_zero] using lt_iff_re_im (z := z) (w := 0)
/-- With `z ≤ w` iff `w - z` is real and nonnegative, `ℝ` and `ℂ` are star ordered rings.
(That is, a star ring in which the nonnegative elements are those of the form `star z * z`.)

Note this is only an instance with `open scoped ComplexOrder`. -/
def toStarOrderedRing : StarOrderedRing K :=
  StarOrderedRing.ofNonnegIff'
    (h_add := fun {x y} hxy z => by
      

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
x y : K
hxy : x ≤ y
z : K
⊢ z + x ≤ z + y

Next tactic:
rw [IsROrC.le_iff_re_im] at *
/-- With `z ≤ w` iff `w - z` is real and nonnegative, `ℝ` and `ℂ` are star ordered rings.
(That is, a star ring in which the nonnegative elements are those of the form `star z * z`.)

Note this is only an instance with `open scoped ComplexOrder`. -/
def toStarOrderedRing : StarOrderedRing K :=
  StarOrderedRing.ofNonnegIff'
    (h_add := fun {x y} hxy z => by
      rw [IsROrC.le_iff_re_im] at *
      

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
x y : K
hxy : re x ≤ re y ∧ im x = im y
z : K
⊢ re (z + x) ≤ re (z + y) ∧ im (z + x) = im (z + y)

Next tactic:
simpa [map_add, add_le_add_iff_left, add_right_inj] using hxy
/-- With `z ≤ w` iff `w - z` is real and nonnegative, `ℝ` and `ℂ` are star ordered rings.
(That is, a star ring in which the nonnegative elements are those of the form `star z * z`.)

Note this is only an instance with `open scoped ComplexOrder`. -/
def toStarOrderedRing : StarOrderedRing K :=
  StarOrderedRing.ofNonnegIff'
    (h_add := fun {x y} hxy z => by
      rw [IsROrC.le_iff_re_im] at *
      simpa [map_add, add_le_add_iff_left, add_right_inj] using hxy)
    (h_nonneg_iff := fun x => by
      

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
x : K
⊢ 0 ≤ x ↔ ∃ s, x = star s * s

Next tactic:
rw [IsROrC.le_iff_re_im, map_zero, map_zero, IsROrC.star_def, eq_comm]
/-- With `z ≤ w` iff `w - z` is real and nonnegative, `ℝ` and `ℂ` are star ordered rings.
(That is, a star ring in which the nonnegative elements are those of the form `star z * z`.)

Note this is only an instance with `open scoped ComplexOrder`. -/
def toStarOrderedRing : StarOrderedRing K :=
  StarOrderedRing.ofNonnegIff'
    (h_add := fun {x y} hxy z => by
      rw [IsROrC.le_iff_re_im] at *
      simpa [map_add, add_le_add_iff_left, add_right_inj] using hxy)
    (h_nonneg_iff := fun x => by
      rw [IsROrC.le_iff_re_im, map_zero, map_zero, IsROrC.star_def, eq_comm]
      

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
x : K
⊢ 0 ≤ re x ∧ im x = 0 ↔ ∃ s, x = (starRingEnd K) s * s

Next tactic:
constructor
/-- With `z ≤ w` iff `w - z` is real and nonnegative, `ℝ` and `ℂ` are star ordered rings.
(That is, a star ring in which the nonnegative elements are those of the form `star z * z`.)

Note this is only an instance with `open scoped ComplexOrder`. -/
def toStarOrderedRing : StarOrderedRing K :=
  StarOrderedRing.ofNonnegIff'
    (h_add := fun {x y} hxy z => by
      rw [IsROrC.le_iff_re_im] at *
      simpa [map_add, add_le_add_iff_left, add_right_inj] using hxy)
    (h_nonneg_iff := fun x => by
      rw [IsROrC.le_iff_re_im, map_zero, map_zero, IsROrC.star_def, eq_comm]
      constructor
      · 

Proof state:
case mp
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
x : K
⊢ 0 ≤ re x ∧ im x = 0 → ∃ s, x = (starRingEnd K) s * s

Next tactic:
rintro ⟨hr, hi⟩
/-- With `z ≤ w` iff `w - z` is real and nonnegative, `ℝ` and `ℂ` are star ordered rings.
(That is, a star ring in which the nonnegative elements are those of the form `star z * z`.)

Note this is only an instance with `open scoped ComplexOrder`. -/
def toStarOrderedRing : StarOrderedRing K :=
  StarOrderedRing.ofNonnegIff'
    (h_add := fun {x y} hxy z => by
      rw [IsROrC.le_iff_re_im] at *
      simpa [map_add, add_le_add_iff_left, add_right_inj] using hxy)
    (h_nonneg_iff := fun x => by
      rw [IsROrC.le_iff_re_im, map_zero, map_zero, IsROrC.star_def, eq_comm]
      constructor
      · rintro ⟨hr, hi⟩
        

Proof state:
case mp.intro
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
x : K
hr : 0 ≤ re x
hi : im x = 0
⊢ ∃ s, x = (starRingEnd K) s * s

Next tactic:
refine ⟨Real.sqrt (IsROrC.re x), ?_⟩
/-- With `z ≤ w` iff `w - z` is real and nonnegative, `ℝ` and `ℂ` are star ordered rings.
(That is, a star ring in which the nonnegative elements are those of the form `star z * z`.)

Note this is only an instance with `open scoped ComplexOrder`. -/
def toStarOrderedRing : StarOrderedRing K :=
  StarOrderedRing.ofNonnegIff'
    (h_add := fun {x y} hxy z => by
      rw [IsROrC.le_iff_re_im] at *
      simpa [map_add, add_le_add_iff_left, add_right_inj] using hxy)
    (h_nonneg_iff := fun x => by
      rw [IsROrC.le_iff_re_im, map_zero, map_zero, IsROrC.star_def, eq_comm]
      constructor
      · rintro ⟨hr, hi⟩
        refine ⟨Real.sqrt (IsROrC.re x), ?_⟩
        

Proof state:
case mp.intro
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
x : K
hr : 0 ≤ re x
hi : im x = 0
⊢ x = (starRingEnd K) ↑(Real.sqrt (re x)) * ↑(Real.sqrt (re x))

Next tactic:
have := (IsROrC.is_real_TFAE x).out 2 3
/-- With `z ≤ w` iff `w - z` is real and nonnegative, `ℝ` and `ℂ` are star ordered rings.
(That is, a star ring in which the nonnegative elements are those of the form `star z * z`.)

Note this is only an instance with `open scoped ComplexOrder`. -/
def toStarOrderedRing : StarOrderedRing K :=
  StarOrderedRing.ofNonnegIff'
    (h_add := fun {x y} hxy z => by
      rw [IsROrC.le_iff_re_im] at *
      simpa [map_add, add_le_add_iff_left, add_right_inj] using hxy)
    (h_nonneg_iff := fun x => by
      rw [IsROrC.le_iff_re_im, map_zero, map_zero, IsROrC.star_def, eq_comm]
      constructor
      · rintro ⟨hr, hi⟩
        refine ⟨Real.sqrt (IsROrC.re x), ?_⟩
        have := (IsROrC.is_real_TFAE x).out 2 3
        

Proof state:
case mp.intro
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
x : K
hr : 0 ≤ re x
hi : im x = 0
this : ↑(re x) = x ↔ im x = 0
⊢ x = (starRingEnd K) ↑(Real.sqrt (re x)) * ↑(Real.sqrt (re x))

Next tactic:
rw [IsROrC.conj_ofReal, ← IsROrC.ofReal_mul, Real.mul_self_sqrt hr, eq_comm, this, hi]
/-- With `z ≤ w` iff `w - z` is real and nonnegative, `ℝ` and `ℂ` are star ordered rings.
(That is, a star ring in which the nonnegative elements are those of the form `star z * z`.)

Note this is only an instance with `open scoped ComplexOrder`. -/
def toStarOrderedRing : StarOrderedRing K :=
  StarOrderedRing.ofNonnegIff'
    (h_add := fun {x y} hxy z => by
      rw [IsROrC.le_iff_re_im] at *
      simpa [map_add, add_le_add_iff_left, add_right_inj] using hxy)
    (h_nonneg_iff := fun x => by
      rw [IsROrC.le_iff_re_im, map_zero, map_zero, IsROrC.star_def, eq_comm]
      constructor
      · rintro ⟨hr, hi⟩
        refine ⟨Real.sqrt (IsROrC.re x), ?_⟩
        have := (IsROrC.is_real_TFAE x).out 2 3
        rw [IsROrC.conj_ofReal, ← IsROrC.ofReal_mul, Real.mul_self_sqrt hr, eq_comm, this, hi]
      · 

Proof state:
case mpr
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
x : K
⊢ (∃ s, x = (starRingEnd K) s * s) → 0 ≤ re x ∧ im x = 0

Next tactic:
rintro ⟨s, rfl⟩
/-- With `z ≤ w` iff `w - z` is real and nonnegative, `ℝ` and `ℂ` are star ordered rings.
(That is, a star ring in which the nonnegative elements are those of the form `star z * z`.)

Note this is only an instance with `open scoped ComplexOrder`. -/
def toStarOrderedRing : StarOrderedRing K :=
  StarOrderedRing.ofNonnegIff'
    (h_add := fun {x y} hxy z => by
      rw [IsROrC.le_iff_re_im] at *
      simpa [map_add, add_le_add_iff_left, add_right_inj] using hxy)
    (h_nonneg_iff := fun x => by
      rw [IsROrC.le_iff_re_im, map_zero, map_zero, IsROrC.star_def, eq_comm]
      constructor
      · rintro ⟨hr, hi⟩
        refine ⟨Real.sqrt (IsROrC.re x), ?_⟩
        have := (IsROrC.is_real_TFAE x).out 2 3
        rw [IsROrC.conj_ofReal, ← IsROrC.ofReal_mul, Real.mul_self_sqrt hr, eq_comm, this, hi]
      · rintro ⟨s, rfl⟩
        

Proof state:
case mpr.intro
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
s : K
⊢ 0 ≤ re ((starRingEnd K) s * s) ∧ im ((starRingEnd K) s * s) = 0

Next tactic:
simp only [IsROrC.star_def, IsROrC.conj_mul]
/-- With `z ≤ w` iff `w - z` is real and nonnegative, `ℝ` and `ℂ` are star ordered rings.
(That is, a star ring in which the nonnegative elements are those of the form `star z * z`.)

Note this is only an instance with `open scoped ComplexOrder`. -/
def toStarOrderedRing : StarOrderedRing K :=
  StarOrderedRing.ofNonnegIff'
    (h_add := fun {x y} hxy z => by
      rw [IsROrC.le_iff_re_im] at *
      simpa [map_add, add_le_add_iff_left, add_right_inj] using hxy)
    (h_nonneg_iff := fun x => by
      rw [IsROrC.le_iff_re_im, map_zero, map_zero, IsROrC.star_def, eq_comm]
      constructor
      · rintro ⟨hr, hi⟩
        refine ⟨Real.sqrt (IsROrC.re x), ?_⟩
        have := (IsROrC.is_real_TFAE x).out 2 3
        rw [IsROrC.conj_ofReal, ← IsROrC.ofReal_mul, Real.mul_self_sqrt hr, eq_comm, this, hi]
      · rintro ⟨s, rfl⟩
        simp only [IsROrC.star_def, IsROrC.conj_mul]
        

Proof state:
case mpr.intro
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
s : K
⊢ 0 ≤ re ↑(normSq s) ∧ im ↑(normSq s) = 0

Next tactic:
rw [IsROrC.ofReal_re, IsROrC.ofReal_im, eq_self, and_true]
/-- With `z ≤ w` iff `w - z` is real and nonnegative, `ℝ` and `ℂ` are star ordered rings.
(That is, a star ring in which the nonnegative elements are those of the form `star z * z`.)

Note this is only an instance with `open scoped ComplexOrder`. -/
def toStarOrderedRing : StarOrderedRing K :=
  StarOrderedRing.ofNonnegIff'
    (h_add := fun {x y} hxy z => by
      rw [IsROrC.le_iff_re_im] at *
      simpa [map_add, add_le_add_iff_left, add_right_inj] using hxy)
    (h_nonneg_iff := fun x => by
      rw [IsROrC.le_iff_re_im, map_zero, map_zero, IsROrC.star_def, eq_comm]
      constructor
      · rintro ⟨hr, hi⟩
        refine ⟨Real.sqrt (IsROrC.re x), ?_⟩
        have := (IsROrC.is_real_TFAE x).out 2 3
        rw [IsROrC.conj_ofReal, ← IsROrC.ofReal_mul, Real.mul_self_sqrt hr, eq_comm, this, hi]
      · rintro ⟨s, rfl⟩
        simp only [IsROrC.star_def, IsROrC.conj_mul]
        rw [IsROrC.ofReal_re, IsROrC.ofReal_im, eq_self, and_true]
        

Proof state:
case mpr.intro
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
s : K
⊢ 0 ≤ normSq s

Next tactic:
apply IsROrC.normSq_nonneg
/-- With `z ≤ w` iff `w - z` is real and nonnegative, `ℝ` and `ℂ` are strictly ordered rings.

Note this is only an instance with `open scoped ComplexOrder`. -/
def toStrictOrderedCommRing : StrictOrderedCommRing K where
  zero_le_one := by 

Proof state:
K : Type u_1
E : Type u_2
inst✝ : IsROrC K
⊢ 0 ≤ 1

Next tactic:
simp [@IsROrC.le_iff_re_im K]