l3lab/ntp-mathlib huggingface dataset:
theorem innerProp_neg_one : innerProp' E ((-1 : ℤ) : 𝕜) := by
  intro x y
  simp only [inner_, neg_mul_eq_neg_mul, one_mul, Int.cast_one, one_smul, RingHom.map_one, map_neg,
    Int.cast_neg, neg_smul, neg_one_mul]
  rw [neg_mul_comm]
  congr 1
  have h₁ : ‖-x - y‖ = ‖x + y‖ := by rw [← neg_add', norm_neg]
  have h₂ : ‖-x + y‖ = ‖x - y‖ := by rw [← neg_sub, norm_neg, sub_eq_neg_add]
  have h₃ : ‖(I : 𝕜) • -x + y‖ = ‖(I : 𝕜) • x - y‖ := by
    rw [← neg_sub, norm_neg, sub_eq_neg_add, ← smul_neg]
  

Proof state:
case e_a
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
h₁ : ‖-x - y‖ = ‖x + y‖
h₂ : ‖-x + y‖ = ‖x - y‖
h₃ : ‖I • -x + y‖ = ‖I • x - y‖
⊢ 𝓚 ‖-x + y‖ * 𝓚 ‖-x + y‖ - 𝓚 ‖-x - y‖ * 𝓚 ‖-x - y‖ + I * 𝓚 ‖I • -x + y‖ * 𝓚 ‖I • -x + y‖ -
      I * 𝓚 ‖I • -x - y‖ * 𝓚 ‖I • -x - y‖ =
    -(𝓚 ‖x + y‖ * 𝓚 ‖x + y‖ - 𝓚 ‖x - y‖ * 𝓚 ‖x - y‖ + I * 𝓚 ‖I • x + y‖ * 𝓚 ‖I • x + y‖ -
        I * 𝓚 ‖I • x - y‖ * 𝓚 ‖I • x - y‖)

Next tactic:
have h₄ : ‖(I : 𝕜) • -x - y‖ = ‖(I : 𝕜) • x + y‖ := by rw [smul_neg, ← neg_add', norm_neg]
theorem innerProp_neg_one : innerProp' E ((-1 : ℤ) : 𝕜) := by
  intro x y
  simp only [inner_, neg_mul_eq_neg_mul, one_mul, Int.cast_one, one_smul, RingHom.map_one, map_neg,
    Int.cast_neg, neg_smul, neg_one_mul]
  rw [neg_mul_comm]
  congr 1
  have h₁ : ‖-x - y‖ = ‖x + y‖ := by rw [← neg_add', norm_neg]
  have h₂ : ‖-x + y‖ = ‖x - y‖ := by rw [← neg_sub, norm_neg, sub_eq_neg_add]
  have h₃ : ‖(I : 𝕜) • -x + y‖ = ‖(I : 𝕜) • x - y‖ := by
    rw [← neg_sub, norm_neg, sub_eq_neg_add, ← smul_neg]
  have h₄ : ‖(I : 𝕜) • -x - y‖ = ‖(I : 𝕜) • x + y‖ := by 

Proof state:
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
h₁ : ‖-x - y‖ = ‖x + y‖
h₂ : ‖-x + y‖ = ‖x - y‖
h₃ : ‖I • -x + y‖ = ‖I • x - y‖
⊢ ‖I • -x - y‖ = ‖I • x + y‖

Next tactic:
rw [smul_neg, ← neg_add', norm_neg]
theorem innerProp_neg_one : innerProp' E ((-1 : ℤ) : 𝕜) := by
  intro x y
  simp only [inner_, neg_mul_eq_neg_mul, one_mul, Int.cast_one, one_smul, RingHom.map_one, map_neg,
    Int.cast_neg, neg_smul, neg_one_mul]
  rw [neg_mul_comm]
  congr 1
  have h₁ : ‖-x - y‖ = ‖x + y‖ := by rw [← neg_add', norm_neg]
  have h₂ : ‖-x + y‖ = ‖x - y‖ := by rw [← neg_sub, norm_neg, sub_eq_neg_add]
  have h₃ : ‖(I : 𝕜) • -x + y‖ = ‖(I : 𝕜) • x - y‖ := by
    rw [← neg_sub, norm_neg, sub_eq_neg_add, ← smul_neg]
  have h₄ : ‖(I : 𝕜) • -x - y‖ = ‖(I : 𝕜) • x + y‖ := by rw [smul_neg, ← neg_add', norm_neg]
  

Proof state:
case e_a
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
h₁ : ‖-x - y‖ = ‖x + y‖
h₂ : ‖-x + y‖ = ‖x - y‖
h₃ : ‖I • -x + y‖ = ‖I • x - y‖
h₄ : ‖I • -x - y‖ = ‖I • x + y‖
⊢ 𝓚 ‖-x + y‖ * 𝓚 ‖-x + y‖ - 𝓚 ‖-x - y‖ * 𝓚 ‖-x - y‖ + I * 𝓚 ‖I • -x + y‖ * 𝓚 ‖I • -x + y‖ -
      I * 𝓚 ‖I • -x - y‖ * 𝓚 ‖I • -x - y‖ =
    -(𝓚 ‖x + y‖ * 𝓚 ‖x + y‖ - 𝓚 ‖x - y‖ * 𝓚 ‖x - y‖ + I * 𝓚 ‖I • x + y‖ * 𝓚 ‖I • x + y‖ -
        I * 𝓚 ‖I • x - y‖ * 𝓚 ‖I • x - y‖)

Next tactic:
rw [h₁, h₂, h₃, h₄]
theorem innerProp_neg_one : innerProp' E ((-1 : ℤ) : 𝕜) := by
  intro x y
  simp only [inner_, neg_mul_eq_neg_mul, one_mul, Int.cast_one, one_smul, RingHom.map_one, map_neg,
    Int.cast_neg, neg_smul, neg_one_mul]
  rw [neg_mul_comm]
  congr 1
  have h₁ : ‖-x - y‖ = ‖x + y‖ := by rw [← neg_add', norm_neg]
  have h₂ : ‖-x + y‖ = ‖x - y‖ := by rw [← neg_sub, norm_neg, sub_eq_neg_add]
  have h₃ : ‖(I : 𝕜) • -x + y‖ = ‖(I : 𝕜) • x - y‖ := by
    rw [← neg_sub, norm_neg, sub_eq_neg_add, ← smul_neg]
  have h₄ : ‖(I : 𝕜) • -x - y‖ = ‖(I : 𝕜) • x + y‖ := by rw [smul_neg, ← neg_add', norm_neg]
  rw [h₁, h₂, h₃, h₄]
  

Proof state:
case e_a
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
h₁ : ‖-x - y‖ = ‖x + y‖
h₂ : ‖-x + y‖ = ‖x - y‖
h₃ : ‖I • -x + y‖ = ‖I • x - y‖
h₄ : ‖I • -x - y‖ = ‖I • x + y‖
⊢ 𝓚 ‖x - y‖ * 𝓚 ‖x - y‖ - 𝓚 ‖x + y‖ * 𝓚 ‖x + y‖ + I * 𝓚 ‖I • x - y‖ * 𝓚 ‖I • x - y‖ -
      I * 𝓚 ‖I • x + y‖ * 𝓚 ‖I • x + y‖ =
    -(𝓚 ‖x + y‖ * 𝓚 ‖x + y‖ - 𝓚 ‖x - y‖ * 𝓚 ‖x - y‖ + I * 𝓚 ‖I • x + y‖ * 𝓚 ‖I • x + y‖ -
        I * 𝓚 ‖I • x - y‖ * 𝓚 ‖I • x - y‖)

Next tactic:
ring
theorem _root_.Continuous.inner_ {f g : ℝ → E} (hf : Continuous f) (hg : Continuous g) :
    Continuous fun x => inner_ 𝕜 (f x) (g x) := by
  

Proof state:
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f g : ℝ → E
hf : Continuous f
hg : Continuous g
⊢ Continuous fun x => _root_.inner_ 𝕜 (f x) (g x)

Next tactic:
unfold inner_
theorem _root_.Continuous.inner_ {f g : ℝ → E} (hf : Continuous f) (hg : Continuous g) :
    Continuous fun x => inner_ 𝕜 (f x) (g x) := by
  unfold inner_
  

Proof state:
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f g : ℝ → E
hf : Continuous f
hg : Continuous g
⊢ Continuous fun x =>
    4⁻¹ *
      (𝓚 ‖f x + g x‖ * 𝓚 ‖f x + g x‖ - 𝓚 ‖f x - g x‖ * 𝓚 ‖f x - g x‖ + I * 𝓚 ‖I • f x + g x‖ * 𝓚 ‖I • f x + g x‖ -
        I * 𝓚 ‖I • f x - g x‖ * 𝓚 ‖I • f x - g x‖)

Next tactic:
have := Continuous.const_smul (M := 𝕜) hf I
theorem _root_.Continuous.inner_ {f g : ℝ → E} (hf : Continuous f) (hg : Continuous g) :
    Continuous fun x => inner_ 𝕜 (f x) (g x) := by
  unfold inner_
  have := Continuous.const_smul (M := 𝕜) hf I
  

Proof state:
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f g : ℝ → E
hf : Continuous f
hg : Continuous g
this : Continuous fun x => I • f x
⊢ Continuous fun x =>
    4⁻¹ *
      (𝓚 ‖f x + g x‖ * 𝓚 ‖f x + g x‖ - 𝓚 ‖f x - g x‖ * 𝓚 ‖f x - g x‖ + I * 𝓚 ‖I • f x + g x‖ * 𝓚 ‖I • f x + g x‖ -
        I * 𝓚 ‖I • f x - g x‖ * 𝓚 ‖I • f x - g x‖)

Next tactic:
continuity
theorem inner_.norm_sq (x : E) : ‖x‖ ^ 2 = re (inner_ 𝕜 x x) := by
  

Proof state:
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
⊢ ‖x‖ ^ 2 = re (inner_ 𝕜 x x)

Next tactic:
simp only [inner_]
theorem inner_.norm_sq (x : E) : ‖x‖ ^ 2 = re (inner_ 𝕜 x x) := by
  simp only [inner_]
  

Proof state:
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
⊢ ‖x‖ ^ 2 =
    re
      (4⁻¹ *
        (𝓚 ‖x + x‖ * 𝓚 ‖x + x‖ - 𝓚 ‖x - x‖ * 𝓚 ‖x - x‖ + I * 𝓚 ‖I • x + x‖ * 𝓚 ‖I • x + x‖ -
          I * 𝓚 ‖I • x - x‖ * 𝓚 ‖I • x - x‖))

Next tactic:
have h₁ : IsROrC.normSq (4 : 𝕜) = 16 := by
    have : ((4 : ℝ) : 𝕜) = (4 : 𝕜) := by norm_cast
    rw [← this, normSq_eq_def', IsROrC.norm_of_nonneg (by norm_num : (0 : ℝ) ≤ 4)]
    norm_num
theorem inner_.norm_sq (x : E) : ‖x‖ ^ 2 = re (inner_ 𝕜 x x) := by
  simp only [inner_]
  have h₁ : IsROrC.normSq (4 : 𝕜) = 16 := by
    

Proof state:
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
⊢ normSq 4 = 16

Next tactic:
have : ((4 : ℝ) : 𝕜) = (4 : 𝕜) := by norm_cast
theorem inner_.norm_sq (x : E) : ‖x‖ ^ 2 = re (inner_ 𝕜 x x) := by
  simp only [inner_]
  have h₁ : IsROrC.normSq (4 : 𝕜) = 16 := by
    have : ((4 : ℝ) : 𝕜) = (4 : 𝕜) := by 

Proof state:
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
⊢ ↑4 = 4

Next tactic:
norm_cast
theorem inner_.norm_sq (x : E) : ‖x‖ ^ 2 = re (inner_ 𝕜 x x) := by
  simp only [inner_]
  have h₁ : IsROrC.normSq (4 : 𝕜) = 16 := by
    have : ((4 : ℝ) : 𝕜) = (4 : 𝕜) := by norm_cast
    

Proof state:
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
this : ↑4 = 4
⊢ normSq 4 = 16

Next tactic:
rw [← this, normSq_eq_def', IsROrC.norm_of_nonneg (by norm_num : (0 : ℝ) ≤ 4)]
theorem inner_.norm_sq (x : E) : ‖x‖ ^ 2 = re (inner_ 𝕜 x x) := by
  simp only [inner_]
  have h₁ : IsROrC.normSq (4 : 𝕜) = 16 := by
    have : ((4 : ℝ) : 𝕜) = (4 : 𝕜) := by norm_cast
    rw [← this, normSq_eq_def', IsROrC.norm_of_nonneg (by 

Proof state:
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
this : ↑4 = 4
⊢ 0 ≤ 4

Next tactic:
norm_num
theorem inner_.norm_sq (x : E) : ‖x‖ ^ 2 = re (inner_ 𝕜 x x) := by
  simp only [inner_]
  have h₁ : IsROrC.normSq (4 : 𝕜) = 16 := by
    have : ((4 : ℝ) : 𝕜) = (4 : 𝕜) := by norm_cast
    rw [← this, normSq_eq_def', IsROrC.norm_of_nonneg (by norm_num : (0 : ℝ) ≤ 4)]
    

Proof state:
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
this : ↑4 = 4
⊢ 4 ^ 2 = 16

Next tactic:
norm_num
theorem inner_.norm_sq (x : E) : ‖x‖ ^ 2 = re (inner_ 𝕜 x x) := by
  simp only [inner_]
  have h₁ : IsROrC.normSq (4 : 𝕜) = 16 := by
    have : ((4 : ℝ) : 𝕜) = (4 : 𝕜) := by norm_cast
    rw [← this, normSq_eq_def', IsROrC.norm_of_nonneg (by norm_num : (0 : ℝ) ≤ 4)]
    norm_num
  

Proof state:
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
h₁ : normSq 4 = 16
⊢ ‖x‖ ^ 2 =
    re
      (4⁻¹ *
        (𝓚 ‖x + x‖ * 𝓚 ‖x + x‖ - 𝓚 ‖x - x‖ * 𝓚 ‖x - x‖ + I * 𝓚 ‖I • x + x‖ * 𝓚 ‖I • x + x‖ -
          I * 𝓚 ‖I • x - x‖ * 𝓚 ‖I • x - x‖))

Next tactic:
have h₂ : ‖x + x‖ = 2 * ‖x‖ := by rw [← two_smul 𝕜, norm_smul, IsROrC.norm_two]
theorem inner_.norm_sq (x : E) : ‖x‖ ^ 2 = re (inner_ 𝕜 x x) := by
  simp only [inner_]
  have h₁ : IsROrC.normSq (4 : 𝕜) = 16 := by
    have : ((4 : ℝ) : 𝕜) = (4 : 𝕜) := by norm_cast
    rw [← this, normSq_eq_def', IsROrC.norm_of_nonneg (by norm_num : (0 : ℝ) ≤ 4)]
    norm_num
  have h₂ : ‖x + x‖ = 2 * ‖x‖ := by 

Proof state:
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
h₁ : normSq 4 = 16
⊢ ‖x + x‖ = 2 * ‖x‖

Next tactic:
rw [← two_smul 𝕜, norm_smul, IsROrC.norm_two]
theorem inner_.norm_sq (x : E) : ‖x‖ ^ 2 = re (inner_ 𝕜 x x) := by
  simp only [inner_]
  have h₁ : IsROrC.normSq (4 : 𝕜) = 16 := by
    have : ((4 : ℝ) : 𝕜) = (4 : 𝕜) := by norm_cast
    rw [← this, normSq_eq_def', IsROrC.norm_of_nonneg (by norm_num : (0 : ℝ) ≤ 4)]
    norm_num
  have h₂ : ‖x + x‖ = 2 * ‖x‖ := by rw [← two_smul 𝕜, norm_smul, IsROrC.norm_two]
  

Proof state:
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
h₁ : normSq 4 = 16
h₂ : ‖x + x‖ = 2 * ‖x‖
⊢ ‖x‖ ^ 2 =
    re
      (4⁻¹ *
        (𝓚 ‖x + x‖ * 𝓚 ‖x + x‖ - 𝓚 ‖x - x‖ * 𝓚 ‖x - x‖ + I * 𝓚 ‖I • x + x‖ * 𝓚 ‖I • x + x‖ -
          I * 𝓚 ‖I • x - x‖ * 𝓚 ‖I • x - x‖))

Next tactic:
simp only [h₁, h₂, algebraMap_eq_ofReal, sub_self, norm_zero, mul_re, inv_re, ofNat_re, map_sub,
    map_add, ofReal_re, ofNat_im, ofReal_im, mul_im, I_re, inv_im]
theorem inner_.norm_sq (x : E) : ‖x‖ ^ 2 = re (inner_ 𝕜 x x) := by
  simp only [inner_]
  have h₁ : IsROrC.normSq (4 : 𝕜) = 16 := by
    have : ((4 : ℝ) : 𝕜) = (4 : 𝕜) := by norm_cast
    rw [← this, normSq_eq_def', IsROrC.norm_of_nonneg (by norm_num : (0 : ℝ) ≤ 4)]
    norm_num
  have h₂ : ‖x + x‖ = 2 * ‖x‖ := by rw [← two_smul 𝕜, norm_smul, IsROrC.norm_two]
  simp only [h₁, h₂, algebraMap_eq_ofReal, sub_self, norm_zero, mul_re, inv_re, ofNat_re, map_sub,
    map_add, ofReal_re, ofNat_im, ofReal_im, mul_im, I_re, inv_im]
  

Proof state:
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
h₁ : normSq 4 = 16
h₂ : ‖x + x‖ = 2 * ‖x‖
⊢ ‖x‖ ^ 2 =
    4 / 16 *
        (2 * ‖x‖ * (2 * ‖x‖) - 0 * 0 - 0 +
            ((0 * ‖I • x + x‖ - im I * 0) * ‖I • x + x‖ - (0 * 0 + im I * ‖I • x + x‖) * 0) -
          ((0 * ‖I • x - x‖ - im I * 0) * ‖I • x - x‖ - (0 * 0 + im I * ‖I • x - x‖) * 0)) -
      -0 / 16 *
        (2 * ‖x‖ * 0 + 0 * (2 * ‖x‖) - (0 * 0 + 0 * 0) +
            ((0 * ‖I • x + x‖ - im I * 0) * 0 + (0 * 0 + im I * ‖I • x + x‖) * ‖I • x + x‖) -
          ((0 * ‖I • x - x‖ - im I * 0) * 0 + (0 * 0 + im I * ‖I • x - x‖) * ‖I • x - x‖))

Next tactic:
ring
theorem inner_.conj_symm (x y : E) : conj (inner_ 𝕜 y x) = inner_ 𝕜 x y := by
  

Proof state:
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
⊢ (starRingEnd 𝕜) (inner_ 𝕜 y x) = inner_ 𝕜 x y

Next tactic:
simp only [inner_]
theorem inner_.conj_symm (x y : E) : conj (inner_ 𝕜 y x) = inner_ 𝕜 x y := by
  simp only [inner_]
  

Proof state:
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
⊢ (starRingEnd 𝕜)
      (4⁻¹ *
        (𝓚 ‖y + x‖ * 𝓚 ‖y + x‖ - 𝓚 ‖y - x‖ * 𝓚 ‖y - x‖ + I * 𝓚 ‖I • y + x‖ * 𝓚 ‖I • y + x‖ -
          I * 𝓚 ‖I • y - x‖ * 𝓚 ‖I • y - x‖)) =
    4⁻¹ *
      (𝓚 ‖x + y‖ * 𝓚 ‖x + y‖ - 𝓚 ‖x - y‖ * 𝓚 ‖x - y‖ + I * 𝓚 ‖I • x + y‖ * 𝓚 ‖I • x + y‖ -
        I * 𝓚 ‖I • x - y‖ * 𝓚 ‖I • x - y‖)

Next tactic:
have h4 : conj (4⁻¹ : 𝕜) = 4⁻¹ := by norm_num
theorem inner_.conj_symm (x y : E) : conj (inner_ 𝕜 y x) = inner_ 𝕜 x y := by
  simp only [inner_]
  have h4 : conj (4⁻¹ : 𝕜) = 4⁻¹ := by 

Proof state:
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
⊢ (starRingEnd 𝕜) 4⁻¹ = 4⁻¹

Next tactic:
norm_num
theorem inner_.conj_symm (x y : E) : conj (inner_ 𝕜 y x) = inner_ 𝕜 x y := by
  simp only [inner_]
  have h4 : conj (4⁻¹ : 𝕜) = 4⁻¹ := by norm_num
  

Proof state:
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
h4 : (starRingEnd 𝕜) 4⁻¹ = 4⁻¹
⊢ (starRingEnd 𝕜)
      (4⁻¹ *
        (𝓚 ‖y + x‖ * 𝓚 ‖y + x‖ - 𝓚 ‖y - x‖ * 𝓚 ‖y - x‖ + I * 𝓚 ‖I • y + x‖ * 𝓚 ‖I • y + x‖ -
          I * 𝓚 ‖I • y - x‖ * 𝓚 ‖I • y - x‖)) =
    4⁻¹ *
      (𝓚 ‖x + y‖ * 𝓚 ‖x + y‖ - 𝓚 ‖x - y‖ * 𝓚 ‖x - y‖ + I * 𝓚 ‖I • x + y‖ * 𝓚 ‖I • x + y‖ -
        I * 𝓚 ‖I • x - y‖ * 𝓚 ‖I • x - y‖)

Next tactic:
rw [map_mul, h4]
theorem inner_.conj_symm (x y : E) : conj (inner_ 𝕜 y x) = inner_ 𝕜 x y := by
  simp only [inner_]
  have h4 : conj (4⁻¹ : 𝕜) = 4⁻¹ := by norm_num
  rw [map_mul, h4]
  

Proof state:
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
h4 : (starRingEnd 𝕜) 4⁻¹ = 4⁻¹
⊢ 4⁻¹ *
      (starRingEnd 𝕜)
        (𝓚 ‖y + x‖ * 𝓚 ‖y + x‖ - 𝓚 ‖y - x‖ * 𝓚 ‖y - x‖ + I * 𝓚 ‖I • y + x‖ * 𝓚 ‖I • y + x‖ -
          I * 𝓚 ‖I • y - x‖ * 𝓚 ‖I • y - x‖) =
    4⁻¹ *
      (𝓚 ‖x + y‖ * 𝓚 ‖x + y‖ - 𝓚 ‖x - y‖ * 𝓚 ‖x - y‖ + I * 𝓚 ‖I • x + y‖ * 𝓚 ‖I • x + y‖ -
        I * 𝓚 ‖I • x - y‖ * 𝓚 ‖I • x - y‖)

Next tactic:
congr 1
theorem inner_.conj_symm (x y : E) : conj (inner_ 𝕜 y x) = inner_ 𝕜 x y := by
  simp only [inner_]
  have h4 : conj (4⁻¹ : 𝕜) = 4⁻¹ := by norm_num
  rw [map_mul, h4]
  congr 1
  

Proof state:
case e_a
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
h4 : (starRingEnd 𝕜) 4⁻¹ = 4⁻¹
⊢ (starRingEnd 𝕜)
      (𝓚 ‖y + x‖ * 𝓚 ‖y + x‖ - 𝓚 ‖y - x‖ * 𝓚 ‖y - x‖ + I * 𝓚 ‖I • y + x‖ * 𝓚 ‖I • y + x‖ -
        I * 𝓚 ‖I • y - x‖ * 𝓚 ‖I • y - x‖) =
    𝓚 ‖x + y‖ * 𝓚 ‖x + y‖ - 𝓚 ‖x - y‖ * 𝓚 ‖x - y‖ + I * 𝓚 ‖I • x + y‖ * 𝓚 ‖I • x + y‖ -
      I * 𝓚 ‖I • x - y‖ * 𝓚 ‖I • x - y‖

Next tactic:
simp only [map_sub, map_add, algebraMap_eq_ofReal, ← ofReal_mul, conj_ofReal, map_mul, conj_I]
theorem inner_.conj_symm (x y : E) : conj (inner_ 𝕜 y x) = inner_ 𝕜 x y := by
  simp only [inner_]
  have h4 : conj (4⁻¹ : 𝕜) = 4⁻¹ := by norm_num
  rw [map_mul, h4]
  congr 1
  simp only [map_sub, map_add, algebraMap_eq_ofReal, ← ofReal_mul, conj_ofReal, map_mul, conj_I]
  

Proof state:
case e_a
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
h4 : (starRingEnd 𝕜) 4⁻¹ = 4⁻¹
⊢ ↑(‖y + x‖ * ‖y + x‖) - ↑(‖y - x‖ * ‖y - x‖) + -I * ↑‖I • y + x‖ * ↑‖I • y + x‖ - -I * ↑‖I • y - x‖ * ↑‖I • y - x‖ =
    ↑(‖x + y‖ * ‖x + y‖) - ↑(‖x - y‖ * ‖x - y‖) + I * ↑‖I • x + y‖ * ↑‖I • x + y‖ - I * ↑‖I • x - y‖ * ↑‖I • x - y‖

Next tactic:
rw [add_comm y x, norm_sub_rev]
theorem inner_.conj_symm (x y : E) : conj (inner_ 𝕜 y x) = inner_ 𝕜 x y := by
  simp only [inner_]
  have h4 : conj (4⁻¹ : 𝕜) = 4⁻¹ := by norm_num
  rw [map_mul, h4]
  congr 1
  simp only [map_sub, map_add, algebraMap_eq_ofReal, ← ofReal_mul, conj_ofReal, map_mul, conj_I]
  rw [add_comm y x, norm_sub_rev]
  

Proof state:
case e_a
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
h4 : (starRingEnd 𝕜) 4⁻¹ = 4⁻¹
⊢ ↑(‖x + y‖ * ‖x + y‖) - ↑(‖x - y‖ * ‖x - y‖) + -I * ↑‖I • y + x‖ * ↑‖I • y + x‖ - -I * ↑‖I • y - x‖ * ↑‖I • y - x‖ =
    ↑(‖x + y‖ * ‖x + y‖) - ↑(‖x - y‖ * ‖x - y‖) + I * ↑‖I • x + y‖ * ↑‖I • x + y‖ - I * ↑‖I • x - y‖ * ↑‖I • x - y‖

Next tactic:
by_cases hI : (I : 𝕜) = 0
theorem inner_.conj_symm (x y : E) : conj (inner_ 𝕜 y x) = inner_ 𝕜 x y := by
  simp only [inner_]
  have h4 : conj (4⁻¹ : 𝕜) = 4⁻¹ := by norm_num
  rw [map_mul, h4]
  congr 1
  simp only [map_sub, map_add, algebraMap_eq_ofReal, ← ofReal_mul, conj_ofReal, map_mul, conj_I]
  rw [add_comm y x, norm_sub_rev]
  by_cases hI : (I : 𝕜) = 0
  · 

Proof state:
case pos
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
h4 : (starRingEnd 𝕜) 4⁻¹ = 4⁻¹
hI : I = 0
⊢ ↑(‖x + y‖ * ‖x + y‖) - ↑(‖x - y‖ * ‖x - y‖) + -I * ↑‖I • y + x‖ * ↑‖I • y + x‖ - -I * ↑‖I • y - x‖ * ↑‖I • y - x‖ =
    ↑(‖x + y‖ * ‖x + y‖) - ↑(‖x - y‖ * ‖x - y‖) + I * ↑‖I • x + y‖ * ↑‖I • x + y‖ - I * ↑‖I • x - y‖ * ↑‖I • x - y‖

Next tactic:
simp only [hI, neg_zero, zero_mul]
theorem inner_.conj_symm (x y : E) : conj (inner_ 𝕜 y x) = inner_ 𝕜 x y := by
  simp only [inner_]
  have h4 : conj (4⁻¹ : 𝕜) = 4⁻¹ := by norm_num
  rw [map_mul, h4]
  congr 1
  simp only [map_sub, map_add, algebraMap_eq_ofReal, ← ofReal_mul, conj_ofReal, map_mul, conj_I]
  rw [add_comm y x, norm_sub_rev]
  by_cases hI : (I : 𝕜) = 0
  · simp only [hI, neg_zero, zero_mul]
  -- Porting note: this replaces `norm_I_of_ne_zero` which does not exist in Lean 4
  

Proof state:
case neg
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
h4 : (starRingEnd 𝕜) 4⁻¹ = 4⁻¹
hI : ¬I = 0
⊢ ↑(‖x + y‖ * ‖x + y‖) - ↑(‖x - y‖ * ‖x - y‖) + -I * ↑‖I • y + x‖ * ↑‖I • y + x‖ - -I * ↑‖I • y - x‖ * ↑‖I • y - x‖ =
    ↑(‖x + y‖ * ‖x + y‖) - ↑(‖x - y‖ * ‖x - y‖) + I * ↑‖I • x + y‖ * ↑‖I • x + y‖ - I * ↑‖I • x - y‖ * ↑‖I • x - y‖

Next tactic:
have : ‖(I : 𝕜)‖ = 1 := by
    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 inner_.conj_symm (x y : E) : conj (inner_ 𝕜 y x) = inner_ 𝕜 x y := by
  simp only [inner_]
  have h4 : conj (4⁻¹ : 𝕜) = 4⁻¹ := by norm_num
  rw [map_mul, h4]
  congr 1
  simp only [map_sub, map_add, algebraMap_eq_ofReal, ← ofReal_mul, conj_ofReal, map_mul, conj_I]
  rw [add_comm y x, norm_sub_rev]
  by_cases hI : (I : 𝕜) = 0
  · simp only [hI, neg_zero, zero_mul]
  -- Porting note: this replaces `norm_I_of_ne_zero` which does not exist in Lean 4
  have : ‖(I : 𝕜)‖ = 1 := by
    

Proof state:
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
h4 : (starRingEnd 𝕜) 4⁻¹ = 4⁻¹
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 inner_.conj_symm (x y : E) : conj (inner_ 𝕜 y x) = inner_ 𝕜 x y := by
  simp only [inner_]
  have h4 : conj (4⁻¹ : 𝕜) = 4⁻¹ := by norm_num
  rw [map_mul, h4]
  congr 1
  simp only [map_sub, map_add, algebraMap_eq_ofReal, ← ofReal_mul, conj_ofReal, map_mul, conj_I]
  rw [add_comm y x, norm_sub_rev]
  by_cases hI : (I : 𝕜) = 0
  · simp only [hI, neg_zero, zero_mul]
  -- Porting note: this replaces `norm_I_of_ne_zero` which does not exist in Lean 4
  have : ‖(I : 𝕜)‖ = 1 := by
    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]
  

Proof state:
case neg
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
h4 : (starRingEnd 𝕜) 4⁻¹ = 4⁻¹
hI : ¬I = 0
this : ‖I‖ = 1
⊢ ↑(‖x + y‖ * ‖x + y‖) - ↑(‖x - y‖ * ‖x - y‖) + -I * ↑‖I • y + x‖ * ↑‖I • y + x‖ - -I * ↑‖I • y - x‖ * ↑‖I • y - x‖ =
    ↑(‖x + y‖ * ‖x + y‖) - ↑(‖x - y‖ * ‖x - y‖) + I * ↑‖I • x + y‖ * ↑‖I • x + y‖ - I * ↑‖I • x - y‖ * ↑‖I • x - y‖

Next tactic:
have h₁ : ‖(I : 𝕜) • y - x‖ = ‖(I : 𝕜) • x + y‖ := by
    trans ‖(I : 𝕜) • ((I : 𝕜) • y - x)‖
    · rw [norm_smul, this, one_mul]
    · rw [smul_sub, smul_smul, I_mul_I_of_nonzero hI, neg_one_smul, ← neg_add', add_comm, norm_neg]
theorem inner_.conj_symm (x y : E) : conj (inner_ 𝕜 y x) = inner_ 𝕜 x y := by
  simp only [inner_]
  have h4 : conj (4⁻¹ : 𝕜) = 4⁻¹ := by norm_num
  rw [map_mul, h4]
  congr 1
  simp only [map_sub, map_add, algebraMap_eq_ofReal, ← ofReal_mul, conj_ofReal, map_mul, conj_I]
  rw [add_comm y x, norm_sub_rev]
  by_cases hI : (I : 𝕜) = 0
  · simp only [hI, neg_zero, zero_mul]
  -- Porting note: this replaces `norm_I_of_ne_zero` which does not exist in Lean 4
  have : ‖(I : 𝕜)‖ = 1 := by
    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]
  have h₁ : ‖(I : 𝕜) • y - x‖ = ‖(I : 𝕜) • x + y‖ := by
    

Proof state:
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
h4 : (starRingEnd 𝕜) 4⁻¹ = 4⁻¹
hI : ¬I = 0
this : ‖I‖ = 1
⊢ ‖I • y - x‖ = ‖I • x + y‖

Next tactic:
trans ‖(I : 𝕜) • ((I : 𝕜) • y - x)‖
theorem inner_.conj_symm (x y : E) : conj (inner_ 𝕜 y x) = inner_ 𝕜 x y := by
  simp only [inner_]
  have h4 : conj (4⁻¹ : 𝕜) = 4⁻¹ := by norm_num
  rw [map_mul, h4]
  congr 1
  simp only [map_sub, map_add, algebraMap_eq_ofReal, ← ofReal_mul, conj_ofReal, map_mul, conj_I]
  rw [add_comm y x, norm_sub_rev]
  by_cases hI : (I : 𝕜) = 0
  · simp only [hI, neg_zero, zero_mul]
  -- Porting note: this replaces `norm_I_of_ne_zero` which does not exist in Lean 4
  have : ‖(I : 𝕜)‖ = 1 := by
    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]
  have h₁ : ‖(I : 𝕜) • y - x‖ = ‖(I : 𝕜) • x + y‖ := by
    trans ‖(I : 𝕜) • ((I : 𝕜) • y - x)‖
    · 

Proof state:
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
h4 : (starRingEnd 𝕜) 4⁻¹ = 4⁻¹
hI : ¬I = 0
this : ‖I‖ = 1
⊢ ‖I • y - x‖ = ‖I • (I • y - x)‖

Next tactic:
rw [norm_smul, this, one_mul]
theorem inner_.conj_symm (x y : E) : conj (inner_ 𝕜 y x) = inner_ 𝕜 x y := by
  simp only [inner_]
  have h4 : conj (4⁻¹ : 𝕜) = 4⁻¹ := by norm_num
  rw [map_mul, h4]
  congr 1
  simp only [map_sub, map_add, algebraMap_eq_ofReal, ← ofReal_mul, conj_ofReal, map_mul, conj_I]
  rw [add_comm y x, norm_sub_rev]
  by_cases hI : (I : 𝕜) = 0
  · simp only [hI, neg_zero, zero_mul]
  -- Porting note: this replaces `norm_I_of_ne_zero` which does not exist in Lean 4
  have : ‖(I : 𝕜)‖ = 1 := by
    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]
  have h₁ : ‖(I : 𝕜) • y - x‖ = ‖(I : 𝕜) • x + y‖ := by
    trans ‖(I : 𝕜) • ((I : 𝕜) • y - x)‖
    · rw [norm_smul, this, one_mul]
    · 

Proof state:
𝕜 : Type u_1
inst✝² : IsROrC 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x y : E
h4 : (starRingEnd 𝕜) 4⁻¹ = 4⁻¹
hI : ¬I = 0
this : ‖I‖ = 1
⊢ ‖I • (I • y - x)‖ = ‖I • x + y‖

Next tactic:
rw [smul_sub, smul_smul, I_mul_I_of_nonzero hI, neg_one_smul, ← neg_add', add_comm, norm_neg]